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    1011

  11001

_____

100100

There are 24 "different" cars:

(S,B,M)
(S,B,T)
(S,B,C)
(S,B,P)
(S,W,M)
(S,W,T)
(S,W,C)
(S,W,P)
(S,R,M)
(S,R,T)
(S,R,C)
(S,R,P)
(A,B,M)
(A,B,T)
(A,B,C)
(A,B,P)
(A,W,M)
(A,W,T)
(A,W,C)
(A,W,P)
(A,R,M)
(A,R,T)
(A,R,C)
(A,R,P)

Notice that there are 4 ways to pair the Toyotas with a "different" Mercedes-Benz, "different" Chevrolet or "different" Pinto.  So, 4 + 4 + 4 = 12

And there are 4 ways  to pair the Mercedes-Benz with a "different" Chevrolet or "different'' Pinto. {We've already paired each with a "different" Toyota.} So 4 + 4 = 8

And since we've already paired the Chevrolet with a "different" Toyota or Mercedes-Benz, we have 4 ways that we can pair them with a "different" Pinto.

So there are 12 + 8 + 4 = 24 "different" pairings

And the total number of possible pairs made by selecting any 2 cars from 24  = C(24,2) = 276

So the probability is 24 / 276 = 2/23.

Great!  I got the same answer, but I like your method better !

Bruce rate  100m/15 s =  6 2/3 m/s

   Allen was there 3 seconds before Bruce who still had to run   3 sec x 6 2/3 m/s = 20 meters

if each roll is a sequence and you roll 5 times. it should be 5 sequences. maybe 

Well, you can express it in any number of ways:

500 000 000 000 000 000 000 =5E20 or 5 x 10^20 or 6^26.6 .......etc.

a=999;b=1;c=1;s=0;p=0;b=c;cycle:s=s+(b%10);b=int(b/10);if(b!=0, goto cycle, 0);c++;if(c<=a, goto5,0);print"Total Sum =",s;d=int(log(a)+0.1)+1;f=1;c=1;e=(a-c+1)*d;loop:e=e-(9*(d-f)*10^(f-1)); f++;if(f<=d, goto loop, 0);print"Total Num =",e

OUTPUT:

Total Sum = 13500
Total Num = 2889

There are 300 7s between 1 and 999 (counting the repeats such as 77, 777......etc).

So, the probability that a random digit will be a 7 is:

300 / 2889 =~ 10.38 %.

Note: This is NOT a mathematical answer but a computer code generated answer!! Just do simple counting like this: 1+2+3+4......9 =45 + 45 * 9 (for ones digits) + 45 * 10 (for tens digits) =900 between 1 and 99....etc.

For (b), there are 3*9*8 = 216 numbers that have one 7, 3*9 = 27 numbers that have two 7s, and one number that has three 7s, so there are a total of 1*216 + 2*27 + 3*1 = 273 sevens among all the numbers.  Divide this by your answer to part (a), and you'll get the probability.

never mind i solved part a) and it was correct. 

i have no idea how to approach b), i know it probably includes a lot of choosing but i don't know if it's multiplying/adding and i don't know how to figure out which ones differ in all three categories. even a hint would be good. thank you!!

but doesn't that only work when one of the unique beads are at place 1? what about when one of the same colored beads are at place 1?

We can place one of the unique beads, and call it reference number 1.  Then the possible places of the two same colored beads are at 2 and 6, 3 or 5 and 4, so three ways.  The rest of the 4 charms can be placed in 4! ways.  So the number of arrangements is 3*4! = 72.

Thank you:)

ab = 34,  a+b=34,

so,

(a + 1)(b + 1) = ab + a + b + 1 = 34+ 34 +1 = 69.

where did the 9 come from?

Why is it 6*9? I got 6!/2! because there are 6! total possibilities dividing 2! because there are two of the same, but it was incorrect. 

Also 6*9 is 54

There are 6*9 = 45 ways of arranging the beads.

1. They sent the fingerprints to Interpol, who forwarded them to the FBI. The FBI fingerprint examiner found a match. It was a match to a man named Brandon Mayfield, an attorney from Oregon.

2. They say that fingerprint identification is infallible, and that it is 100% accurate. The problems with these assumptions, as shown in the Mayfield case, is that sometimes people mess up.

3. Fingerprint matches are determined through a series of points and loops, whorls, and arches.

4. Fingerprint analysis is different on television because in reality there is an

This is my attempt at this one:

180 = 2 x 2 x 3 x 3 x 5 =15 - This is the sum of the prime factors of 180.

This one has : 5! /2!2! =30 permutations

So, 30 / 120 = 1/4 = 25 % probability 

144 =4 x 2 x 2 x 3 x 3 = 14 - This is  one of the the sums of the prime factors of 144.

144 =2 x 2 x 2 x 3 x 6 = 15 - This is the other possible sum of the prime factors of 144.

The first one has : 5! / 2!2! = 30 permutations

The 2nd one has :  5! / 3!    = 20 permutations

And this one has : [30 + 20] /120 =41.667% probability

Therefore, the product probability of 144 > 180.

P. S. For goodness sake, some mathematician should check this !!! Thanks. 

The roots of x^2 - 34x + 34 = 0      are  17 + sqrt 255     and    17 - sqrt 255

then   (a + 1)(b + 1)   =   (18 + sqrt255)(18-sqrt255)       Can you calculate the answer from here?

The last three digits of \(7^{2020}\) are \(001\)

When doing these sorts of problems, you take the factorial of the total number of letters, then divide it by however many same letters there are. In this case it would be 5!/2!. I divide by the 2! since there are 2 letter o's. Simplifying, we get 120/2 = 60. Hope I helped!

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