We pick $3$ numbers (one by one) from set $\{0,1,...,100\}$. What is probabilty that two numbers are the same if sum of those $3$ numbers is $100$?
My solution: Which two are the same we can pick in $\binom {3}{2}$ ways. Suggest $x_2=x_3$- we need to find compositon $x_1+x_2+x_2=100 \implies x_1+2x_2=100$ which implies that $x_1$ is even so we can divide this by $2$. Now we get $y_1+y_2=50$ , and using formula there are $$\binom{50+2-1}{2-1}=51$$ compositions. So, probability is $$\frac{51*3}{\binom{100+3-1}{3-1}}$$
Is this right answer?
P.S.$\binom{100+3-1}{3-1}$ is number of compositions of 100 into 3 parts (allowing $0$)