Probability of objects are being sorted

Question

There are 20 toffees (10 of mint, 10 of orange) in two boxes. I mix them together. Then I put 10 randomly chosen from the pot into one box and 10 in the other box. What is the probability that I end up with 10 orange toffees in one box and 10 mint toffees in the other?

Answer 1

There is only 2 possible ways to get the desired arrangement. Mint in box A, rest in box B or Orange in box A and mint in box B.

The total number of ways you can randomly put is $\frac {20!} {10! 10!}$ or $20 \choose 10$.

So the probability is $\frac 2 {20 \choose 10}$ or $\frac {2. 10!. 10!} {20!}$

Answer 2

You can view this question via the following process.

First, choose 10 toffees - there are $20 \choose 10$ such choices and put them into one box. Put the remaining toffees in the other box.

There are only two scenarios where you get the desired outcome (perfectly separated), i.e. choosing all 10 orange first, or choosing all 10 mint first, thus you would have:

$$ Pr(\text{Perfectly Separated}) = \frac{2}{20 \choose 10} $$