Confused about multinomials. Can we write $\binom{n}{a,b,c}=\binom{n}{a}\binom{n-a}{b}\binom{n-a-b}{c}$ if $a+b+c \le n$?

Question

Can we write $\binom{n}{a,b,c}=\binom{n}{a}\binom{n-a}{b}\binom{n-a-b}{c}$ if $a+b+c \le n$?

The definition for multinomial says $a+b+c=n$ must hold or else $\binom{n}{a,b,c}=0$.

I found that if $a+b+c \ge n$ we get $\binom{n}{a,b,c}=0$, but if $a+b+c \le n$, then $\binom{n}{a,b,c}=\binom{n}{a,b,c, n-(a+b+c)}$

Answer 1

No, for any $(a,b,c)\in\Bbb N^3$ and $a+b+c\leq n$ then,

$$\dbinom{n}{a}\dbinom{n-a}{b}\dbinom{n-a-b}{c}=\dbinom{n\qquad\qquad\qquad}{a,b,c, n-a-b-c}\;(n-a-b-c)!$$

Unlike the binomial coefficient, the usual convention is not to leave the last lower term of the multinomial coefficient implicit.   The sum of the lower terms is required to equal the upper term.

If you are using non-standard notation you should mention this somewhere (in your work).