How many 10-digit ternary strings are there that contain exactly two 0s, three 1s, and five 2s?

There are ways of picking 2 of the 10 available positions for a 0. 8 positions remain.

There are ways of picking 3 of the 8 available positions for a 1. 5 positions remain, but we're filling all of them with 2s, and there's way of doing that.

So we have

The last expression has a more compact form in terms of the so-called multinomial coefficient,

step-by-step explanation:

probability for j is 1/4

while

a is 1/6