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5.)

What is the probability that a randomly selected three-digit number has the property that one digit is equal to the product of the other two? Express your answer as a common fraction.

To make                            Possible products                  #  of Numbers

0                               2nd, 3rd digits 0..1st digit 1 - 9        9

1                                           11                                        1

2                                           21                                        3

3                                           31                                        3

4                                           14 , 22                                 6                               

5                                           15                                        3

6                                           16   23                                 9                               

7                                            71                                       3

8                                            81   42                                9

9                                            91   33                                6

Prob  =   52 / 900    =   13 / 225

Using    

A little tricky....but......we can imagine the sine graph shifted down  one unit from the normal "parent" function  and having an amplitude of 3

So...far...we have this

y =  3sin (Bx)- 1

Note that a  "starting point"  on the curve is  (0, -1)  and a period of 1.5  puts us at  (-1, 4pi)

So.....  a period must be    4pi / 1.5  =  (8/3) pi

So.... to  find the number of periods in 2pi ,  ("B"), we can solve this

(2pi / (8/3 pi)  = B               

(6/3 pi) / (8/3 pi)  =

(6/3)(8/3)  =

6/8  =   3/4  =  B =  there are 3/4 periods in 2pi

So...our  function is

y  = 3sin [ (3/4)x ] -  1

Here's the graph : https://www.desmos.com/calculator/olu4zazirp

[ Note....other solutions are also possible ....this just seemed to be the easiest ]

So....

2). Of the 5 numbers given[2, 3, 4, 7, 8], we have the following pairs of numbers that each permutation of the 5 numbers must end in order to be divisible by 4. And they are as follows:

24, 28, 32, 48, 72, 84, or 6 pairs. Each of these pairs will cycle through all the permutations of the remaining 3 numbers, or 3!.

Therefore the probability that the resulting number is divisible by 4 will be:

[6 x 3!] / 5! = 36 / 120 =30%.

4.)

A bowl contains only red marbles, blue marbles and green marbles. The probability of selecting a red marble from the bowl is 3/13. The probability of selecting a blue marble from the bowl is 2/5. There are fewer than 100 marbles in the bowl. What is the probability of selecting a green marble and then a red marble from the bowl on the first two selections, assuming marbles are not returned to the bowl after being pulled?

There must be 65 marbles in the bowl because  5  and 13  have the lowest common multple  of  65....

3/13  =  15/65      and    2/5  =  26/65

So  

15 are red,  26 are blue and  65 - 15 - 26  =  24 are green

So...the probability of selecting green and then red  =

(24/65) (15/64)   =   (24/64) (15/65)   =  ( 3/8) (3/13)  =     9 /  104

1.)  There are 3 other aces  and 12 more hearts that will win the small prize

And there are 52 possibilities when we draw a card from the deck.....

So....the probability  of winning a small prize  =

Favorable outcomes  /  Total possible outcomes   =

[ 12 + 3 ] / 52  =

15 / 52

Let the number of adults = A

Let the number of children = C

A + C = 796

4.50A + 3.25C =3,042, solve for A, C

A = 796 - C

4.50[796 - C] + 3.25C =3,042

3,582 - 4.50C + 3.25C = 3,042    subtract 3,582 from both sides

-1.25C = - 540                              divide both sides by -1.25

C = -540 / - 1.25

C = 432 - Number of children

796 - 432 = 364 - Number of adults.

-

Well, trial and error appeared to be my friend here. If I was not aware that the original expression could be simplified further, I probably would have made the same conclusion as you did, Melody. It is considered improper to have radicals or fractional exponents in the denominator, so I knew that there was some way to simplify this. My method is really only relevant to this particular problem. You will see why. 

Now, let's consider that extra bit on the end, the \(9^\frac{1}{3}=\left(3^2\right)^\frac{1}{3}=3^\frac{2}{3}=3^{\left(\frac{1}{3}\right)^2}\). This means that the denominator can be written like \(1+3^\frac{1}{3}+3^{\left(\frac{1}{3}\right)^2}\), and if I let \(x=3^\frac{1}{3}\), then I can represent the denominator like \(x^2+x+1\). I then made a stunning realization. This required some extraordinary observational skills. 

So, no, the conjugate is not "magical." The only issue with this thought process is that this will not help you on most types of problems with multi-term denominators. This is just one particular case, and I happened to crack it. 

Modifying Melody's result a little we have:

1 * * *                     1*4*4*4 = 64 ways, but 4 possible positions for the 1, so 64*4 = 256 ways

1 1 * *                     1*1*4*4 = 16 ways, but 6 possible combinations of the two 1's, so 16*6 = 96 ways

1 1 6 *                     1*1*1*4 = 4  ways, but 6 possible positions for the 1's

and two possible remaining positions for the 6,  so 4*6*2 = 48 ways

1, 1, 1, any              1*1*1*5 = 5 ways, but 4 possible positions for the three 1's, so 5*4 = 20 ways

                                  total   =  256 + 96 + 48 + 20 =  420 ways

Hence probability = 420/1296 → 35/108 ≈ 0.324

. 

Thanks, there could be a 'bug' in my logic too ://

Thanks Melody! The reason I said it was " a lazy person's solution" was because I simply wrote a few lines of computer code and it spat them out in seconds!! It also did the counting from the top!! It could very well have a small bug in it as far as counting is concerned. I might go through it and see if I can spot it!!.

ok so what do you think the correct answer is and  what makes you think it is correct?

AND if you have the working we would like to see that too.  

Please do not just say that any answer is wrong without justifying your remark.

I, and others, would like to learn too.

That is excellent work x^2   BUT

I would like to know what thought processes led you to finding that conjugate.

(Becasue that it the bit that stumped the rest of us)

Hi Guest, you have not done it the lazy way, you have done it the most straight forward (and long) way.

I'll see if I get the same answer...    

All the 720 arrangements of the digits 1 through 6 are put in numerical order. What is the 428th term?

How many are between 100 000 and 199 000   That would be 1*5! = 120

200 000 - 299 000    That would also be 1*5! = 120

300 000 - 399 000    That would also be 1*5! = 120

So that is 360 so far    428-360= 68 more...

So it is going to be between 400,000 and 499,999

How many between 410,000 and 419,999

the position of the 4 and the 1 is set so that would be   4! = 24         68-24= 44 more to go

How many between 420,000 and 429,999. Another 24                     44-24=20 more to go

So it is going to be between 430,000 and 439,999

How many between 431,000 and 431,999.     3!=6

How many between 432,000 and 432,999.     3!=6

We canot repeat the 3 or the 4 at the front so

How many between 435,000 and 435,999.     3!=6                             20-18=2 more to go

Next is 436 125, then 436,215

So I think 436,215 is the 428th term.     That is what I get

check

120+120+120+24+24+6+6+6+2 = 428  Seems ok

they are not the same so one of us has made a mistake  

ab / [ a + b ]   <  a   

Since a,b > 0. multiply both sides by a + b

ab <  a^2 + ab       subtract ab from  both sides

0 <  a^2           which is true

Likewise

ab / [ a + b ]  <  b

ab < ab + b^2

0 < b^2 

Notice that he has   2 options every morning  ( B or S )

So...we should expect

(2) * ( 2)  * (2)    =  8  possible outcomes  

Only the second choice has 8 possible outcomes

If it's even....then

f(-x)  =  f(x)

So  f(-x)  =  2(-x)^3 - (-x)^2    =  -2x^3 - x^2   but this does not =  f(x)

If it's odd....then  f(-x)  = - f(x)

And  - f(x)     =     -[2x^3  - x^2]  =  -2x^3 + x^2    but  this doesn't = f(-x)  

So.....neither 

The function doesn't  "rise" or "fall"  from (2, 6 )  ....so it's constant here

Notice.... that at each light, there are 3 possibilities....

So....the number  of total possibilities is just  (3) * (3) * (3)  =   27

There  are 4 * 4  =  16 possible outcomes

1, 3, 4, 6

Let us find the possibilities where the sum is > 5

(1, 6)    (6, 1)  ( 3, 4) ( 4, 3) (3, 6) (6, 3)  ( 4, 6) ( 6, 4) (3, 3) (4, 4) (6, 6)

So  the ones that include at least one odd number are

(1,6) (6,1), (3,4) (4,3) (3,6) (6,3) (3,3)  =    7

Possible outcomes   =  3 numbers * 4 letters   =   12

The outcomes that show "C"   are  {1,C}, {2, C}  and {4, C}  =  3 outcomes

So........

9 outcomes do not show "C"

Let the cost ot a pencil be x

Let the cost of a notebook be y

Let the cost of an eraser be z 

So

x + y + z  =  1.00   ⇒  z  = 100 - x - y

x > 2z   ⇒  x > 2 (100 -x - y) ⇒ x > 200 - 2x - 2y ⇒ 2y > 200 - 3x  (1)

8z > 3y ⇒ 8(100 - x - y)  > 3y ⇒  800 - 8x - 8y  > 3y  ⇒ 800 - 8x > 11y   (2)    

5y > 6x   (3)

Here is the graph of (1), (2)  and (3) :

https://www.desmos.com/calculator/dfd3tgokb7

The integer solutions occur inside the triangle formed by the three inequalites (feasible region)

One is when  x  = 37, y = 45   ( and z = 18 )    

Proof

37 > 2(18)  ⇒  37 > 36

8(18) > 3(45)  ⇒ 144 > 135

5(45) > 6(37) ⇒  225 > 222

So..with this solution......the cost of a pencil  =  37 cents

(2^1) (2^2) (2^3)......(2^99) (2^100)  =

2^(1 + 2 + 3 + ...+  99 + 100)  =

2^(5050)  =

(2^10)^505

Note that 2^10    ends in 24

And note the pattern of 24

24^5 ends in 24

24^10  ends in 76

24^15  ends in 24  ....etc....

So......  (....24)^[5 (2n - 1) ]    ends   in 24....when n is a positive integer

So

(2^10)^505    =

(.....24)^505  =

(  ...24 ) ^[ 5 (2 * 51 - 1) ]   .......ends in 24

OK, Young Person!! Ready, GO!!.

{{1, 2, 3, 4, 5, 6}, {1, 2, 3, 4, 6, 5}, {1, 2, 3, 5, 4, 6}, {1, 2, 3, 5, 6, 4}, {1, 2, 3, 6, 4, 5}, {1, 2, 3, 6, 5, 4}, {1, 2, 4, 3, 5, 6}, {1, 2, 4, 3, 6, 5}, {1, 2, 4, 5, 3, 6}, {1, 2, 4, 5, 6, 3}, {1, 2, 4, 6, 3, 5}, {1, 2, 4, 6, 5, 3}, {1, 2, 5, 3, 4, 6}, {1, 2, 5, 3, 6, 4}, {1, 2, 5, 4, 3, 6}, {1, 2, 5, 4, 6, 3}, {1, 2, 5, 6, 3, 4}, {1, 2, 5, 6, 4, 3}, {1, 2, 6, 3, 4, 5}, {1, 2, 6, 3, 5, 4}, {1, 2, 6, 4, 3, 5}, {1, 2, 6, 4, 5, 3}, {1, 2, 6, 5, 3, 4}, {1, 2, 6, 5, 4, 3}, {1, 3, 2, 4, 5, 6}, {1, 3, 2, 4, 6, 5}, {1, 3, 2, 5, 4, 6}, {1, 3, 2, 5, 6, 4}, {1, 3, 2, 6, 4, 5}, {1, 3, 2, 6, 5, 4}, {1, 3, 4, 2, 5, 6}, {1, 3, 4, 2, 6, 5}, {1, 3, 4, 5, 2, 6}, {1, 3, 4, 5, 6, 2}, {1, 3, 4, 6, 2, 5}, {1, 3, 4, 6, 5, 2}, {1, 3, 5, 2, 4, 6}, {1, 3, 5, 2, 6, 4}, {1, 3, 5, 4, 2, 6}, {1, 3, 5, 4, 6, 2}, {1, 3, 5, 6, 2, 4}, {1, 3, 5, 6, 4, 2}, {1, 3, 6, 2, 4, 5}, {1, 3, 6, 2, 5, 4}, {1, 3, 6, 4, 2, 5}, {1, 3, 6, 4, 5, 2}, {1, 3, 6, 5, 2, 4}, {1, 3, 6, 5, 4, 2}, {1, 4, 2, 3, 5, 6}, {1, 4, 2, 3, 6, 5}, {1, 4, 2, 5, 3, 6}, {1, 4, 2, 5, 6, 3}, {1, 4, 2, 6, 3, 5}, {1, 4, 2, 6, 5, 3}, {1, 4, 3, 2, 5, 6}, {1, 4, 3, 2, 6, 5}, {1, 4, 3, 5, 2, 6}, {1, 4, 3, 5, 6, 2}, {1, 4, 3, 6, 2, 5}, {1, 4, 3, 6, 5, 2}, {1, 4, 5, 2, 3, 6}, {1, 4, 5, 2, 6, 3}, {1, 4, 5, 3, 2, 6}, {1, 4, 5, 3, 6, 2}, {1, 4, 5, 6, 2, 3}, {1, 4, 5, 6, 3, 2}, {1, 4, 6, 2, 3, 5}, {1, 4, 6, 2, 5, 3}, {1, 4, 6, 3, 2, 5}, {1, 4, 6, 3, 5, 2}, {1, 4, 6, 5, 2, 3}, {1, 4, 6, 5, 3, 2}, {1, 5, 2, 3, 4, 6}, {1, 5, 2, 3, 6, 4}, {1, 5, 2, 4, 3, 6}, {1, 5, 2, 4, 6, 3}, {1, 5, 2, 6, 3, 4}, {1, 5, 2, 6, 4, 3}, {1, 5, 3, 2, 4, 6}, {1, 5, 3, 2, 6, 4}, {1, 5, 3, 4, 2, 6}, {1, 5, 3, 4, 6, 2}, {1, 5, 3, 6, 2, 4}, {1, 5, 3, 6, 4, 2}, {1, 5, 4, 2, 3, 6}, {1, 5, 4, 2, 6, 3}, {1, 5, 4, 3, 2, 6}, {1, 5, 4, 3, 6, 2}, {1, 5, 4, 6, 2, 3}, {1, 5, 4, 6, 3, 2}, {1, 5, 6, 2, 3, 4}, {1, 5, 6, 2, 4, 3}, {1, 5, 6, 3, 2, 4}, {1, 5, 6, 3, 4, 2}, {1, 5, 6, 4, 2, 3}, {1, 5, 6, 4, 3, 2}, {1, 6, 2, 3, 4, 5}, {1, 6, 2, 3, 5, 4}, {1, 6, 2, 4, 3, 5}, {1, 6, 2, 4, 5, 3}, {1, 6, 2, 5, 3, 4}, {1, 6, 2, 5, 4, 3}, {1, 6, 3, 2, 4, 5}, {1, 6, 3, 2, 5, 4}, {1, 6, 3, 4, 2, 5}, {1, 6, 3, 4, 5, 2}, {1, 6, 3, 5, 2, 4}, {1, 6, 3, 5, 4, 2}, {1, 6, 4, 2, 3, 5}, {1, 6, 4, 2, 5, 3}, {1, 6, 4, 3, 2, 5}, {1, 6, 4, 3, 5, 2}, {1, 6, 4, 5, 2, 3}, {1, 6, 4, 5, 3, 2}, {1, 6, 5, 2, 3, 4}, {1, 6, 5, 2, 4, 3}, {1, 6, 5, 3, 2, 4}, {1, 6, 5, 3, 4, 2}, {1, 6, 5, 4, 2, 3}, {1, 6, 5, 4, 3, 2}, {2, 1, 3, 4, 5, 6}, {2, 1, 3, 4, 6, 5}, {2, 1, 3, 5, 4, 6}, {2, 1, 3, 5, 6, 4}, {2, 1, 3, 6, 4, 5}, {2, 1, 3, 6, 5, 4}, {2, 1, 4, 3, 5, 6}, {2, 1, 4, 3, 6, 5}, {2, 1, 4, 5, 3, 6}, {2, 1, 4, 5, 6, 3}, {2, 1, 4, 6, 3, 5}, {2, 1, 4, 6, 5, 3}, {2, 1, 5, 3, 4, 6}, {2, 1, 5, 3, 6, 4}, {2, 1, 5, 4, 3, 6}, {2, 1, 5, 4, 6, 3}, {2, 1, 5, 6, 3, 4}, {2, 1, 5, 6, 4, 3}, {2, 1, 6, 3, 4, 5}, {2, 1, 6, 3, 5, 4}, {2, 1, 6, 4, 3, 5}, {2, 1, 6, 4, 5, 3}, {2, 1, 6, 5, 3, 4}, {2, 1, 6, 5, 4, 3}, {2, 3, 1, 4, 5, 6}, {2, 3, 1, 4, 6, 5}, {2, 3, 1, 5, 4, 6}, {2, 3, 1, 5, 6, 4}, {2, 3, 1, 6, 4, 5}, {2, 3, 1, 6, 5, 4}, {2, 3, 4, 1, 5, 6}, {2, 3, 4, 1, 6, 5}, {2, 3, 4, 5, 1, 6}, {2, 3, 4, 5, 6, 1}, {2, 3, 4, 6, 1, 5}, {2, 3, 4, 6, 5, 1}, {2, 3, 5, 1, 4, 6}, {2, 3, 5, 1, 6, 4}, {2, 3, 5, 4, 1, 6}, {2, 3, 5, 4, 6, 1}, {2, 3, 5, 6, 1, 4}, {2, 3, 5, 6, 4, 1}, {2, 3, 6, 1, 4, 5}, {2, 3, 6, 1, 5, 4}, {2, 3, 6, 4, 1, 5}, {2, 3, 6, 4, 5, 1}, {2, 3, 6, 5, 1, 4}, {2, 3, 6, 5, 4, 1}, {2, 4, 1, 3, 5, 6}, {2, 4, 1, 3, 6, 5}, {2, 4, 1, 5, 3, 6}, {2, 4, 1, 5, 6, 3}, {2, 4, 1, 6, 3, 5}, {2, 4, 1, 6, 5, 3}, {2, 4, 3, 1, 5, 6}, {2, 4, 3, 1, 6, 5}, {2, 4, 3, 5, 1, 6}, {2, 4, 3, 5, 6, 1}, {2, 4, 3, 6, 1, 5}, {2, 4, 3, 6, 5, 1}, {2, 4, 5, 1, 3, 6}, {2, 4, 5, 1, 6, 3}, {2, 4, 5, 3, 1, 6}, {2, 4, 5, 3, 6, 1}, {2, 4, 5, 6, 1, 3}, {2, 4, 5, 6, 3, 1}, {2, 4, 6, 1, 3, 5}, {2, 4, 6, 1, 5, 3}, {2, 4, 6, 3, 1, 5}, {2, 4, 6, 3, 5, 1}, {2, 4, 6, 5, 1, 3}, {2, 4, 6, 5, 3, 1}, {2, 5, 1, 3, 4, 6}, {2, 5, 1, 3, 6, 4}, {2, 5, 1, 4, 3, 6}, {2, 5, 1, 4, 6, 3}, {2, 5, 1, 6, 3, 4}, {2, 5, 1, 6, 4, 3}, {2, 5, 3, 1, 4, 6}, {2, 5, 3, 1, 6, 4}, {2, 5, 3, 4, 1, 6}, {2, 5, 3, 4, 6, 1}, {2, 5, 3, 6, 1, 4}, {2, 5, 3, 6, 4, 1}, {2, 5, 4, 1, 3, 6}, {2, 5, 4, 1, 6, 3}, {2, 5, 4, 3, 1, 6}, {2, 5, 4, 3, 6, 1}, {2, 5, 4, 6, 1, 3}, {2, 5, 4, 6, 3, 1}, {2, 5, 6, 1, 3, 4}, {2, 5, 6, 1, 4, 3}, {2, 5, 6, 3, 1, 4}, {2, 5, 6, 3, 4, 1}, {2, 5, 6, 4, 1, 3}, {2, 5, 6, 4, 3, 1}, {2, 6, 1, 3, 4, 5}, {2, 6, 1, 3, 5, 4}, {2, 6, 1, 4, 3, 5}, {2, 6, 1, 4, 5, 3}, {2, 6, 1, 5, 3, 4}, {2, 6, 1, 5, 4, 3}, {2, 6, 3, 1, 4, 5}, {2, 6, 3, 1, 5, 4}, {2, 6, 3, 4, 1, 5}, {2, 6, 3, 4, 5, 1}, {2, 6, 3, 5, 1, 4}, {2, 6, 3, 5, 4, 1}, {2, 6, 4, 1, 3, 5}, {2, 6, 4, 1, 5, 3}, {2, 6, 4, 3, 1, 5}, {2, 6, 4, 3, 5, 1}, {2, 6, 4, 5, 1, 3}, {2, 6, 4, 5, 3, 1}, {2, 6, 5, 1, 3, 4}, {2, 6, 5, 1, 4, 3}, {2, 6, 5, 3, 1, 4}, {2, 6, 5, 3, 4, 1}, {2, 6, 5, 4, 1, 3}, {2, 6, 5, 4, 3, 1}, {3, 1, 2, 4, 5, 6}, {3, 1, 2, 4, 6, 5}, {3, 1, 2, 5, 4, 6}, {3, 1, 2, 5, 6, 4}, {3, 1, 2, 6, 4, 5}, {3, 1, 2, 6, 5, 4}, {3, 1, 4, 2, 5, 6}, {3, 1, 4, 2, 6, 5}, {3, 1, 4, 5, 2, 6}, {3, 1, 4, 5, 6, 2}, {3, 1, 4, 6, 2, 5}, {3, 1, 4, 6, 5, 2}, {3, 1, 5, 2, 4, 6}, {3, 1, 5, 2, 6, 4}, {3, 1, 5, 4, 2, 6}, {3, 1, 5, 4, 6, 2}, {3, 1, 5, 6, 2, 4}, {3, 1, 5, 6, 4, 2}, {3, 1, 6, 2, 4, 5}, {3, 1, 6, 2, 5, 4}, {3, 1, 6, 4, 2, 5}, {3, 1, 6, 4, 5, 2}, {3, 1, 6, 5, 2, 4}, {3, 1, 6, 5, 4, 2}, {3, 2, 1, 4, 5, 6}, {3, 2, 1, 4, 6, 5}, {3, 2, 1, 5, 4, 6}, {3, 2, 1, 5, 6, 4}, {3, 2, 1, 6, 4, 5}, {3, 2, 1, 6, 5, 4}, {3, 2, 4, 1, 5, 6}, {3, 2, 4, 1, 6, 5}, {3, 2, 4, 5, 1, 6}, {3, 2, 4, 5, 6, 1}, {3, 2, 4, 6, 1, 5}, {3, 2, 4, 6, 5, 1}, {3, 2, 5, 1, 4, 6}, {3, 2, 5, 1, 6, 4}, {3, 2, 5, 4, 1, 6}, {3, 2, 5, 4, 6, 1}, {3, 2, 5, 6, 1, 4}, {3, 2, 5, 6, 4, 1}, {3, 2, 6, 1, 4, 5}, {3, 2, 6, 1, 5, 4}, {3, 2, 6, 4, 1, 5}, {3, 2, 6, 4, 5, 1}, {3, 2, 6, 5, 1, 4}, {3, 2, 6, 5, 4, 1}, {3, 4, 1, 2, 5, 6}, {3, 4, 1, 2, 6, 5}, {3, 4, 1, 5, 2, 6}, {3, 4, 1, 5, 6, 2}, {3, 4, 1, 6, 2, 5}, {3, 4, 1, 6, 5, 2}, {3, 4, 2, 1, 5, 6}, {3, 4, 2, 1, 6, 5}, {3, 4, 2, 5, 1, 6}, {3, 4, 2, 5, 6, 1}, {3, 4, 2, 6, 1, 5}, {3, 4, 2, 6, 5, 1}, {3, 4, 5, 1, 2, 6}, {3, 4, 5, 1, 6, 2}, {3, 4, 5, 2, 1, 6}, {3, 4, 5, 2, 6, 1}, {3, 4, 5, 6, 1, 2}, {3, 4, 5, 6, 2, 1}, {3, 4, 6, 1, 2, 5}, {3, 4, 6, 1, 5, 2}, {3, 4, 6, 2, 1, 5}, {3, 4, 6, 2, 5, 1}, {3, 4, 6, 5, 1, 2}, {3, 4, 6, 5, 2, 1}, {3, 5, 1, 2, 4, 6}, {3, 5, 1, 2, 6, 4}, {3, 5, 1, 4, 2, 6}, {3, 5, 1, 4, 6, 2}, {3, 5, 1, 6, 2, 4}, {3, 5, 1, 6, 4, 2}, {3, 5, 2, 1, 4, 6}, {3, 5, 2, 1, 6, 4}, {3, 5, 2, 4, 1, 6}, {3, 5, 2, 4, 6, 1}, {3, 5, 2, 6, 1, 4}, {3, 5, 2, 6, 4, 1}, {3, 5, 4, 1, 2, 6}, {3, 5, 4, 1, 6, 2}, {3, 5, 4, 2, 1, 6}, {3, 5, 4, 2, 6, 1}, {3, 5, 4, 6, 1, 2}, {3, 5, 4, 6, 2, 1}, {3, 5, 6, 1, 2, 4}, {3, 5, 6, 1, 4, 2}, {3, 5, 6, 2, 1, 4}, {3, 5, 6, 2, 4, 1}, {3, 5, 6, 4, 1, 2}, {3, 5, 6, 4, 2, 1}, {3, 6, 1, 2, 4, 5}, {3, 6, 1, 2, 5, 4}, {3, 6, 1, 4, 2, 5}, {3, 6, 1, 4, 5, 2}, {3, 6, 1, 5, 2, 4}, {3, 6, 1, 5, 4, 2}, {3, 6, 2, 1, 4, 5}, {3, 6, 2, 1, 5, 4}, {3, 6, 2, 4, 1, 5}, {3, 6, 2, 4, 5, 1}, {3, 6, 2, 5, 1, 4}, {3, 6, 2, 5, 4, 1}, {3, 6, 4, 1, 2, 5}, {3, 6, 4, 1, 5, 2}, {3, 6, 4, 2, 1, 5}, {3, 6, 4, 2, 5, 1}, {3, 6, 4, 5, 1, 2}, {3, 6, 4, 5, 2, 1}, {3, 6, 5, 1, 2, 4}, {3, 6, 5, 1, 4, 2}, {3, 6, 5, 2, 1, 4}, {3, 6, 5, 2, 4, 1}, {3, 6, 5, 4, 1, 2}, {3, 6, 5, 4, 2, 1}, {4, 1, 2, 3, 5, 6}, {4, 1, 2, 3, 6, 5}, {4, 1, 2, 5, 3, 6}, {4, 1, 2, 5, 6, 3}, {4, 1, 2, 6, 3, 5}, {4, 1, 2, 6, 5, 3}, {4, 1, 3, 2, 5, 6}, {4, 1, 3, 2, 6, 5}, {4, 1, 3, 5, 2, 6}, {4, 1, 3, 5, 6, 2}, {4, 1, 3, 6, 2, 5}, {4, 1, 3, 6, 5, 2}, {4, 1, 5, 2, 3, 6}, {4, 1, 5, 2, 6, 3}, {4, 1, 5, 3, 2, 6}, {4, 1, 5, 3, 6, 2}, {4, 1, 5, 6, 2, 3}, {4, 1, 5, 6, 3, 2}, {4, 1, 6, 2, 3, 5}, {4, 1, 6, 2, 5, 3}, {4, 1, 6, 3, 2, 5}, {4, 1, 6, 3, 5, 2}, {4, 1, 6, 5, 2, 3}, {4, 1, 6, 5, 3, 2}, {4, 2, 1, 3, 5, 6}, {4, 2, 1, 3, 6, 5}, {4, 2, 1, 5, 3, 6}, {4, 2, 1, 5, 6, 3}, {4, 2, 1, 6, 3, 5}, {4, 2, 1, 6, 5, 3}, {4, 2, 3, 1, 5, 6}, {4, 2, 3, 1, 6, 5}, {4, 2, 3, 5, 1, 6}, {4, 2, 3, 5, 6, 1}, {4, 2, 3, 6, 1, 5}, {4, 2, 3, 6, 5, 1}, {4, 2, 5, 1, 3, 6}, {4, 2, 5, 1, 6, 3}, {4, 2, 5, 3, 1, 6}, {4, 2, 5, 3, 6, 1}, {4, 2, 5, 6, 1, 3}, {4, 2, 5, 6, 3, 1}, {4, 2, 6, 1, 3, 5}, {4, 2, 6, 1, 5, 3}, {4, 2, 6, 3, 1, 5}, {4, 2, 6, 3, 5, 1}, {4, 2, 6, 5, 1, 3}, {4, 2, 6, 5, 3, 1}, {4, 3, 1, 2, 5, 6}, {4, 3, 1, 2, 6, 5}, {4, 3, 1, 5, 2, 6}, {4, 3, 1, 5, 6, 2}, {4, 3, 1, 6, 2, 5}, {4, 3, 1, 6, 5, 2}, {4, 3, 2, 1, 5, 6}, {4, 3, 2, 1, 6, 5}, {4, 3, 2, 5, 1, 6}, {4, 3, 2, 5, 6, 1}, {4, 3, 2, 6, 1, 5}, {4, 3, 2, 6, 5, 1}, {4, 3, 5, 1, 2, 6}, {4, 3, 5, 1, 6, 2}, {4, 3, 5, 2, 1, 6}, {4, 3, 5, 2, 6, 1}, {4, 3, 5, 6, 1, 2}, {4, 3, 5, 6, 2, 1}, {4, 3, 6, 1, 2, 5}, {4, 3, 6, 1, 5, 2}, {4, 3, 6, 2, 1, 5}, {4, 3, 6, 2, 5, 1}, {4, 3, 6, 5, 1, 2}, {4, 3, 6, 5, 2, 1}, {4, 5, 1, 2, 3, 6}, {4, 5, 1, 2, 6, 3}, {4, 5, 1, 3, 2, 6}, {4, 5, 1, 3, 6, 2}, {4, 5, 1, 6, 2, 3}, {4, 5, 1, 6, 3, 2}, {4, 5, 2, 1, 3, 6}, {4, 5, 2, 1, 6, 3}, {4, 5, 2, 3, 1, 6}, {4, 5, 2, 3, 6, 1}, {4, 5, 2, 6, 1, 3}, {4, 5, 2, 6, 3, 1}, {4, 5, 3, 1, 2, 6}, {4, 5, 3, 1, 6, 2}, {4, 5, 3, 2, 1, 6}, {4, 5, 3, 2, 6, 1}, {4, 5, 3, 6, 1, 2}, {4, 5, 3, 6, 2, 1}, {4, 5, 6, 1, 2, 3}, {4, 5, 6, 1, 3, 2}, {4, 5, 6, 2, 1, 3}, {4, 5, 6, 2, 3, 1}, {4, 5, 6, 3, 1, 2}, {4, 5, 6, 3, 2, 1}, {4, 6, 1, 2, 3, 5}, {4, 6, 1, 2, 5, 3}, {4, 6, 1, 3, 2, 5}, {4, 6, 1, 3, 5, 2}, {4, 6, 1, 5, 2, 3}, {4, 6, 1, 5, 3, 2}, {4, 6, 2, 1, 3, 5}, {4, 6, 2, 1, 5, 3}, {4, 6, 2, 3, 1, 5}, {4, 6, 2, 3, 5, 1}, {4, 6, 2, 5, 1, 3}, {4, 6, 2, 5, 3, 1}, {4, 6, 3, 1, 2, 5}, {4, 6, 3, 1, 5, 2}, {4, 6, 3, 2, 1, 5}, {4, 6, 3, 2, 5, 1}, {4, 6, 3, 5, 1, 2}, {4, 6, 3, 5, 2, 1}, {4, 6, 5, 1, 2, 3}, {4, 6, 5, 1, 3, 2}, {4, 6, 5, 2, 1, 3}, {4, 6, 5, 2, 3, 1}, {4, 6, 5, 3, 1, 2}, {4, 6, 5, 3, 2, 1}, {5, 1, 2, 3, 4, 6}, {5, 1, 2, 3, 6, 4}, {5, 1, 2, 4, 3, 6}, {5, 1, 2, 4, 6, 3}, {5, 1, 2, 6, 3, 4}, {5, 1, 2, 6, 4, 3}, {5, 1, 3, 2, 4, 6}, {5, 1, 3, 2, 6, 4}, {5, 1, 3, 4, 2, 6}, {5, 1, 3, 4, 6, 2}, {5, 1, 3, 6, 2, 4}, {5, 1, 3, 6, 4, 2}, {5, 1, 4, 2, 3, 6}, {5, 1, 4, 2, 6, 3}, {5, 1, 4, 3, 2, 6}, {5, 1, 4, 3, 6, 2}, {5, 1, 4, 6, 2, 3}, {5, 1, 4, 6, 3, 2}, {5, 1, 6, 2, 3, 4}, {5, 1, 6, 2, 4, 3}, {5, 1, 6, 3, 2, 4}, {5, 1, 6, 3, 4, 2}, {5, 1, 6, 4, 2, 3}, {5, 1, 6, 4, 3, 2}, {5, 2, 1, 3, 4, 6}, {5, 2, 1, 3, 6, 4}, {5, 2, 1, 4, 3, 6}, {5, 2, 1, 4, 6, 3}, {5, 2, 1, 6, 3, 4}, {5, 2, 1, 6, 4, 3}, {5, 2, 3, 1, 4, 6}, {5, 2, 3, 1, 6, 4}, {5, 2, 3, 4, 1, 6}, {5, 2, 3, 4, 6, 1}, {5, 2, 3, 6, 1, 4}, {5, 2, 3, 6, 4, 1}, {5, 2, 4, 1, 3, 6}, {5, 2, 4, 1, 6, 3}, {5, 2, 4, 3, 1, 6}, {5, 2, 4, 3, 6, 1}, {5, 2, 4, 6, 1, 3}, {5, 2, 4, 6, 3, 1}, {5, 2, 6, 1, 3, 4}, {5, 2, 6, 1, 4, 3}, {5, 2, 6, 3, 1, 4}, {5, 2, 6, 3, 4, 1}, {5, 2, 6, 4, 1, 3}, {5, 2, 6, 4, 3, 1}, {5, 3, 1, 2, 4, 6}, {5, 3, 1, 2, 6, 4}, {5, 3, 1, 4, 2, 6}, {5, 3, 1, 4, 6, 2}, {5, 3, 1, 6, 2, 4}, {5, 3, 1, 6, 4, 2}, {5, 3, 2, 1, 4, 6}, {5, 3, 2, 1, 6, 4}, {5, 3, 2, 4, 1, 6}, {5, 3, 2, 4, 6, 1}, {5, 3, 2, 6, 1, 4}, {5, 3, 2, 6, 4, 1}, {5, 3, 4, 1, 2, 6}, {5, 3, 4, 1, 6, 2}, {5, 3, 4, 2, 1, 6}, {5, 3, 4, 2, 6, 1}, {5, 3, 4, 6, 1, 2}, {5, 3, 4, 6, 2, 1}, {5, 3, 6, 1, 2, 4}, {5, 3, 6, 1, 4, 2}, {5, 3, 6, 2, 1, 4}, {5, 3, 6, 2, 4, 1}, {5, 3, 6, 4, 1, 2}, {5, 3, 6, 4, 2, 1}, {5, 4, 1, 2, 3, 6}, {5, 4, 1, 2, 6, 3}, {5, 4, 1, 3, 2, 6}, {5, 4, 1, 3, 6, 2}, {5, 4, 1, 6, 2, 3}, {5, 4, 1, 6, 3, 2}, {5, 4, 2, 1, 3, 6}, {5, 4, 2, 1, 6, 3}, {5, 4, 2, 3, 1, 6}, {5, 4, 2, 3, 6, 1}, {5, 4, 2, 6, 1, 3}, {5, 4, 2, 6, 3, 1}, {5, 4, 3, 1, 2, 6}, {5, 4, 3, 1, 6, 2}, {5, 4, 3, 2, 1, 6}, {5, 4, 3, 2, 6, 1}, {5, 4, 3, 6, 1, 2}, {5, 4, 3, 6, 2, 1}, {5, 4, 6, 1, 2, 3}, {5, 4, 6, 1, 3, 2}, {5, 4, 6, 2, 1, 3}, {5, 4, 6, 2, 3, 1}, {5, 4, 6, 3, 1, 2}, {5, 4, 6, 3, 2, 1}, {5, 6, 1, 2, 3, 4}, {5, 6, 1, 2, 4, 3}, {5, 6, 1, 3, 2, 4}, {5, 6, 1, 3, 4, 2}, {5, 6, 1, 4, 2, 3}, {5, 6, 1, 4, 3, 2}, {5, 6, 2, 1, 3, 4}, {5, 6, 2, 1, 4, 3}, {5, 6, 2, 3, 1, 4}, {5, 6, 2, 3, 4, 1}, {5, 6, 2, 4, 1, 3}, {5, 6, 2, 4, 3, 1}, {5, 6, 3, 1, 2, 4}, {5, 6, 3, 1, 4, 2}, {5, 6, 3, 2, 1, 4}, {5, 6, 3, 2, 4, 1}, {5, 6, 3, 4, 1, 2}, {5, 6, 3, 4, 2, 1}, {5, 6, 4, 1, 2, 3}, {5, 6, 4, 1, 3, 2}, {5, 6, 4, 2, 1, 3}, {5, 6, 4, 2, 3, 1}, {5, 6, 4, 3, 1, 2}, {5, 6, 4, 3, 2, 1}, {6, 1, 2, 3, 4, 5}, {6, 1, 2, 3, 5, 4}, {6, 1, 2, 4, 3, 5}, {6, 1, 2, 4, 5, 3}, {6, 1, 2, 5, 3, 4}, {6, 1, 2, 5, 4, 3}, {6, 1, 3, 2, 4, 5}, {6, 1, 3, 2, 5, 4}, {6, 1, 3, 4, 2, 5}, {6, 1, 3, 4, 5, 2}, {6, 1, 3, 5, 2, 4}, {6, 1, 3, 5, 4, 2}, {6, 1, 4, 2, 3, 5}, {6, 1, 4, 2, 5, 3}, {6, 1, 4, 3, 2, 5}, {6, 1, 4, 3, 5, 2}, {6, 1, 4, 5, 2, 3}, {6, 1, 4, 5, 3, 2}, {6, 1, 5, 2, 3, 4}, {6, 1, 5, 2, 4, 3}, {6, 1, 5, 3, 2, 4}, {6, 1, 5, 3, 4, 2}, {6, 1, 5, 4, 2, 3}, {6, 1, 5, 4, 3, 2}, {6, 2, 1, 3, 4, 5}, {6, 2, 1, 3, 5, 4}, {6, 2, 1, 4, 3, 5}, {6, 2, 1, 4, 5, 3}, {6, 2, 1, 5, 3, 4}, {6, 2, 1, 5, 4, 3}, {6, 2, 3, 1, 4, 5}, {6, 2, 3, 1, 5, 4}, {6, 2, 3, 4, 1, 5}, {6, 2, 3, 4, 5, 1}, {6, 2, 3, 5, 1, 4}, {6, 2, 3, 5, 4, 1}, {6, 2, 4, 1, 3, 5}, {6, 2, 4, 1, 5, 3}, {6, 2, 4, 3, 1, 5}, {6, 2, 4, 3, 5, 1}, {6, 2, 4, 5, 1, 3}, {6, 2, 4, 5, 3, 1}, {6, 2, 5, 1, 3, 4}, {6, 2, 5, 1, 4, 3}, {6, 2, 5, 3, 1, 4}, {6, 2, 5, 3, 4, 1}, {6, 2, 5, 4, 1, 3}, {6, 2, 5, 4, 3, 1}, {6, 3, 1, 2, 4, 5}, {6, 3, 1, 2, 5, 4}, {6, 3, 1, 4, 2, 5}, {6, 3, 1, 4, 5, 2}, {6, 3, 1, 5, 2, 4}, {6, 3, 1, 5, 4, 2}, {6, 3, 2, 1, 4, 5}, {6, 3, 2, 1, 5, 4}, {6, 3, 2, 4, 1, 5}, {6, 3, 2, 4, 5, 1}, {6, 3, 2, 5, 1, 4}, {6, 3, 2, 5, 4, 1}, {6, 3, 4, 1, 2, 5}, {6, 3, 4, 1, 5, 2}, {6, 3, 4, 2, 1, 5}, {6, 3, 4, 2, 5, 1}, {6, 3, 4, 5, 1, 2}, {6, 3, 4, 5, 2, 1}, {6, 3, 5, 1, 2, 4}, {6, 3, 5, 1, 4, 2}, {6, 3, 5, 2, 1, 4}, {6, 3, 5, 2, 4, 1}, {6, 3, 5, 4, 1, 2}, {6, 3, 5, 4, 2, 1}, {6, 4, 1, 2, 3, 5}, {6, 4, 1, 2, 5, 3}, {6, 4, 1, 3, 2, 5}, {6, 4, 1, 3, 5, 2}, {6, 4, 1, 5, 2, 3}, {6, 4, 1, 5, 3, 2}, {6, 4, 2, 1, 3, 5}, {6, 4, 2, 1, 5, 3}, {6, 4, 2, 3, 1, 5}, {6, 4, 2, 3, 5, 1}, {6, 4, 2, 5, 1, 3}, {6, 4, 2, 5, 3, 1}, {6, 4, 3, 1, 2, 5}, {6, 4, 3, 1, 5, 2}, {6, 4, 3, 2, 1, 5}, {6, 4, 3, 2, 5, 1}, {6, 4, 3, 5, 1, 2}, {6, 4, 3, 5, 2, 1}, {6, 4, 5, 1, 2, 3}, {6, 4, 5, 1, 3, 2}, {6, 4, 5, 2, 1, 3}, {6, 4, 5, 2, 3, 1}, {6, 4, 5, 3, 1, 2}, {6, 4, 5, 3, 2, 1}, {6, 5, 1, 2, 3, 4}, {6, 5, 1, 2, 4, 3}, {6, 5, 1, 3, 2, 4}, {6, 5, 1, 3, 4, 2}, {6, 5, 1, 4, 2, 3}, {6, 5, 1, 4, 3, 2}, {6, 5, 2, 1, 3, 4}, {6, 5, 2, 1, 4, 3}, {6, 5, 2, 3, 1, 4}, {6, 5, 2, 3, 4, 1}, {6, 5, 2, 4, 1, 3}, {6, 5, 2, 4, 3, 1}, {6, 5, 3, 1, 2, 4}, {6, 5, 3, 1, 4, 2}, {6, 5, 3, 2, 1, 4}, {6, 5, 3, 2, 4, 1}, {6, 5, 3, 4, 1, 2}, {6, 5, 3, 4, 2, 1}, {6, 5, 4, 1, 2, 3}, {6, 5, 4, 1, 3, 2}, {6, 5, 4, 2, 1, 3}, {6, 5, 4, 2, 3, 1}, {6, 5, 4, 3, 1, 2}, {6, 5, 4, 3, 2, 1}}

Note: This is a lazy person's solution !!! If I made a mistake, count them from the top !!!!!　

OK.......

Call the total number of fruits started with, F

So......

After  the seller sells   121 fruits... he has  F - 121 reamaining

And he sells  (1/ 4)  of these in the afternoon =(1/4)(F - 121)

So.....what he/she strated with  - what was  sold in the morning - what was sold in the afternoon = (1/5)F

So....we have this equation

F  -  121  -  (1/4)(F - 121)  =  (1/5)F      simplify

F - 121  - (1/4)F +   121/4 =  (1/5)F

(3/4)F - 121 +  121/4  = (1/5)F

(3/4)F  -  484/4 + 121/4  =  (1/5)F

(3/4)F - 363/4  = (1/5)F       subtract   (1/5)F from both sides..add 363/4 to both sides

(3/4)F - (1/5)F   =  363/4      get a common denominator on the right

(15/20) - (4/20) F  =  363/4

(11/20)F  =  363/4        multiply both sides by   (20/11)

F   =  (363/4) (20/11)  =   (363/11) (20/4)   =  (33)(5)   =   165 fruits

Proof

165  - 121  =     44 left  after the morning

(1/4) of these sold in the afternoon  =  11 sold   and 33 remain......and this is (1/5) of the total

I think your logic is not correct, unfortuantely. If you have heard the story, you have not necessarily "related the story to others." Let's say, for example, that there were 20 storytellers. Since each storyteller tells 2 people, there are 40 people who have been told the story. Now, 60 know about the story, only 20 of which are now storytellers, or "parents."

When you solved and you got 671, you found the number of people who know the story to be 671, but the prompt explains that only 335 people know the story. If I am understanding the problem correctly, 335 should go in for the number of people who know the story, or the "Number of vertices"

\(335=2x+1\\ 334=2x\\ 167=x\)

167 equals the number of parents, or the number of storytellers. 

Sum of arithmetic series from 1 to 12 =[F+L] /2 x N =[1+12] / 2 x 12 =78 combinations of 13 to 24

So, the probability is =78 /(12^2) =78 / 144.