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A bag contains $n$ balls, some of which are white, the others are black, white being more numerous than black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the two balls are of the same color is the same as the probability that they are of different colors. It is given that $180 < n < 220$. If $k$ denotes the number of white balls, find the exact value of $k+n$.

I got $$\binom{k}{2} + \binom{n-k}{2} =2 \binom{k}{1}\binom{n-k}{1}$$

But now how to further solve it

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    $\begingroup$ Have you considered actually writing out the binomial expressions? They are quite simple. $\endgroup$ – Alex R. Apr 24 '17 at 16:47
  • $\begingroup$ Please be consistent between lower case and capital letters. $k$ and $K$ are different. Often they are related variables, but here it appears you intend them to be the same. $\endgroup$ – Ross Millikan Apr 24 '17 at 16:48
  • $\begingroup$ @RossMillikan I just updated the question, converting all to lower case $\endgroup$ – gt6989b Apr 24 '17 at 16:49
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Simplify. Remember than $$\binom{m}{i} = \frac{m!}{i!(m-i)!}$$ and thus $\binom{m}{1} = m$. What is $\binom{m}{2}$? can you plug the results in your equation and solve?

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  • $\begingroup$ After doing all that also . I am not getting . $\endgroup$ – hey Apr 24 '17 at 16:48
  • $\begingroup$ @hey what are you getting? simplify left- and right-hand sides, what do you get when you plug in? $\endgroup$ – gt6989b Apr 24 '17 at 16:49
  • $\begingroup$ $2k^2+N^2-N=4Nk-4k^2$ $\endgroup$ – hey Apr 24 '17 at 16:50
  • $\begingroup$ @hey You should get a quadratic equation in $n,k$. Choose one variable to solve for using the quadratic formula. The stuff under the square root needs to be an integer. $\endgroup$ – Ross Millikan Apr 24 '17 at 16:51
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You will get this:

$$\binom{k}{2} + \binom{n-k}{2} = \binom{k}{1}\binom{n-k}{1}$$

$$so, k(k-1) +(n-k)(n-k-1) = k(n-k)$$ Because you only have exactly k white balls and n-k black balls, from which you have to choose one ball each from the two groups.

After this, you can proceed as @ross-millikan suggested. Solve the equation for k or n and select suitable n in thr domain.

Solving for which you get k = 105, n =196

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