The following was an exercise I solved:
We have 8 numbered balls - two blue, two red, two green, and two yellow. When dividing them between 4 children, 2 balls each, what is the probability at least one child will get two balls of the same color?
I solved it using the exclusion-inclusion principle, and got the end result $3/7$. My question is, since this is such a nice fraction, was there any way of solving the problem such that I could've arrived at the fraction directly?
(This is the calculation I did: http://www.wolframalpha.com/input/?i=C(4%2c1)*(4*C(6%2c2)*C(4%2c2)C(2%2c2))%2f2520-C(4%2c2)(4*3*C(4%2c2)C(2%2c2))%2f2520%2bC(4%2c3)(4*3*2*C(2%2c2))%2f2520-C(4%2c4)*(4*3*2*1)%2f2520&incParTime=true)