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It is exactly the same as my answer without the little error.

I accept that you found it independently though.

It was a   'thank you for answering'     that I was looking for. 

A further comment would have been good too.

If you had bothered to look at imy answer hard enough to find my little error that would have gone down really well.

We do not answer these ONLY to amuse ourselves.

Anyway, I am glad that you know how to do them now.

We have   C(13.2)  ways of choosing the two ranks

And in each of these ranks we have    C(4,2)   ways of choosing the  pair within a  rank

And we  have   C(44,1)  ways of choosing the remaining card

So

C(13,2)  * [ C(4,2) ]^2  * C(44,1)    =

123552 ways

This is impossible guest.....

10 + 13   =   23

But  23  isn't one of the pair  sums....

Four distinct integers a, b, c and d have the property that when added in pairs, the sums 16, 19, 20, 21, 22, and 25 are obtained. What are the four integers in increasing order? (place a comma and then a space between each integer)

Let the integers, in increasing order, be

a, b , c , d

We know the the  sum of the  smallest two must  be  the least sum...so...

a + b   = 16     (1)

And we know that the sum   of the  largest two must be the  greatest sum....so....

c + d  =  25      (2)

And it must be that  

b + d  =   22      (3)

Subtracting  (3)  from (2)    we have that

c - b  =   3       so

b + 3   = c

So

a + b + c + d  = 41

a + b + (b + 3) + d = 41

a + 2b + d  = 38

a + d  =   38  - 2b      (4)    

But   a + d   must be even   because  the right side of  (4)  is even

So  either

a + d  = 20       or      a  +  d   =  22  

If  a + d   = 20 , then  by (4)  ....... b  = 9     c    =   12      d  = 13     and.....by (1)..... a  = 7

This  must be true

a = 7     b   = 9      c   =  12       d   = 13

Check

a + b = 16

a + c = 19

a + d  = 20

b + c  = 21

b + d =  22

c + d =  25

Let h be the third altitude.  Then by the triangle inequality, h + 12 > 14, h + 14 > 12, and 12 + 14 > h.  The largest integer h that satisfies these inequalities is h = 25.

The four integers are 6, 10, 12, 13.

I'm sorry I was posting the answer I found another way as a reply to your answer.

there are 52 ways of choosing the first card

there are 3 ways of chooing its pair

there are 48 ways of choosing the next number

and 3 ways of choosing its pair

there are 44 ways of choosing the single card

so that is    52*3*48*3*44 = 988416

is it 100, it is posted like this: $10#$10, thats like 10 or something!

(100  original subscribers  +  30 new members -  10 that cancelled )  *  ( $10)  =

(120) * ($10)  =

$1200  this month

(3x) ( 5x^4)^2  =

(3x) * ( 5^2)  *  ( x^4)^2  =

(3x ) *  ( 25)  * (x^8)  =

(3 * 25) * ( x * x^8)   =

75x^9 

Using the Law of Sines   ih the lower left  blue triangle, we have

sin  (a)  / s  =  sin (60) / S       (1) 

Where  a  is the angle formed by the side of the yellow triangle  and the base of this triangle

And S is the  side of the  yellow triangle

And s  is the  side of the orange triangle with an area of 20

And using the Law of Sines  in  the lower right blue triangle, we have

sin ( 120 - a)  / (1.5s)  =  sin (60) / S     (2)

Where  (180 - 60 - a)  =   (120 - a)   is the angle formed  by  the side of the  yellow triangle  and the  base of this triangle

And S is the side of the yellow triangle

And 1.5s  is the side of the red triangle

So....equating (1) and (2), we have

sin (a) / s   =    sin (120 - a) / (1.5 s)

sin (a)    =   sin (120 - a) / (1.5)

sin (a)  =  [sin(120) cos(a)  -  sin (a)cos (120) ]  / (3/2)

(3/2)sin (a)   = (√3 / 2 )cos (a)  +   sin (a) (1/2)

3sin (a)  = √3 cos (a)  +  sin a

2sin (a)  =  √3 cos (a)

sin (a) / cos(a)  =  √3/2

tan (a)  = √3/2

So  sin (a)   =    √ 3  / [ √ [ 3 + 4 ]  =  √3 / √7

And we  can find s   as

20  = (√3/4)s^2

80 / √3  = s^2

s  = √ [ 80 / √3  ] 

And we can find the side  of the yellow triangle as 

s / sin (a)  = S  / sin (60)

√ [ 80 / √3  ]  / [ √3 / √7 ] =   S / [√3/2 ]

S =  (√3/2) * √ [ 80 / √3  ]  / [ √3 / √7 ]  =      2√35 / [⁴√3 ]

So....the area  of the  yellow equilateral  triangle is

(√3 / 4)S^2  =

(√3/ 4)  *  (2√35)^2 / √3  =

(1/4) 140   =

140 / 4  =

35  units^2

Thanks Guest #5

Atroshus is also rude for not acknowledging my answer.

There are 45 ways. There is a formula for it, if you want to know, please ask me.

i got the same answer as mathproblem solver101 9z(z^2-3z+3)

to the guest above, at least say where she is incorrect. Its kind of rude to talk like that. In fact, its very rude.

Just found the answer,

Let z=a+bi and \(\bar{z}=a-bi\).

so \(4i\bar{z}=4b+4ia\).

We now find that \(3z+4i\bar{z} = (3a+4b) + (4a+3b)i\)

so \(3a+4b=1\) and \(4a+3b=-8\)

Solving this leads us to a=-5 and b=4

So the answer is \(z=\boxed{-5+4i}\)

Got it double thx!

OK....welcome  !!!

[This is just the "formula"  for  the sum of a finite geometric series   ]

1.10 because 100+10 but since its percent, 1.10

1.10

Just divide: 887! / (886! + 1) =887 - The GCD - accurate to 1 part in 2,231.

Got it

Thank you CPhill!

But it doesn't have a common-

oh is it 10?