I understand what nCr is as far as what the formula is, but I ran across this notation and have no clue how I’m supposed to read this. Is it asking for 3 different combinations (the sum C each letter individually) or is it asking for something else?
$$\binom{i + j + k}{i, j, k}$$
The notation $$\binom ab$$ is used to denote the number of ways to choose a subset of $b$ objects from $a$ distinguishable objects. It can also be thought of as the way to partition a set of $a$ objects into two buckets, the first of which must have $b$ objects and the second of which has $b-a$ objects. This can be seen from its evaluation $$\binom ab=\frac{a!}{b!(a-b)!}.$$ What if you want to count the number of ways to partition a set of $a$ objects into more than $2$ buckets? Say we want a notation to partition a set of size $n_1+n_2+\cdots+n_m$ into $m$ buckets, the first of which is of size $n_1$, the second of which is of size $n_2$, et cetera. This can be represented by the fraction $$\frac{(n_1+n_2+\cdots+n_m)!}{n_1!n_2!\cdots n_m!}.$$ When discussing such numbers, it can be useful to have some shorthand. So, this is often written $$\binom{n_1+n_2+\cdots+n_m}{n_1,n_2,\dots,n_m}.$$ For example, $$\binom8{2,2,4}=\frac{8!}{2!2!4!}=420.$$