easy Probability

Room Divider

A release-checked easy problem for training Combinations and Permutations, Multinomial Coefficient.

Question

There are $n$ people in a room, of which $1 \leq r \leq n$ are special. The people are divided up into $r$ distinct groups of sizes $n_1,\dots,n_r$, with each $n_i \geq 1$. Find the probability that each group consists of one special person when $n = 10, r = 3, n_1 = 2, n_2 = 3,$ and $n_3 = 5$.

Practice focus

This Probability problem is tagged Combinations and Permutations, Multinomial Coefficient. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.