joint PMF in a class of n students

Question

A class of n students takes a test in which each student gets an A with probability p, a B with probability q, and a grade below B with probability 1 − p − q, independently of any other student. If X and Y are the numbers of students that get an A and a B respectively, calculate the joint PMF pX,Y .

I found this answer online:
Let r = 1 − p − q
Then,
Prob( X = i, Y = j) = n!/(i!j!(n-i-j)!)(q^j)(r^(n-i-…....so on) for j=0,1,2..., j=0,1, 2..., 0<=i+j<=n

However I'm not able to figure out how to get to this answer. Any help is appreciated!

Answer 1

The corresponding distribution is known as multinomial distribution.

The rationale is as follows:

Consider we want the first $i$ people to get A, followed by the next $j$ people to get B and the rest get the other grades. The corresponding probability is $p^iq^jr^{n-i-j} $.

However, the original problem doesn't specify the first $i$ must get A and the next $j$ to get B. We just have to pick $i $ of them to get A, $j$ of them to get B. Hence, we multiply the term by $\frac{n!}{i!j! (n-i-j)!} $.