Multinomial identity

#multinomial#identity#coefficient

Statement

Show that $\sum_{i+j+k=n}\frac{n!}{i!j!k!}=3^n$ .

Hint

This is the multinomial theorem with $(1+1+1)^n$ .

Show solutionHide solution

Multinomial. $(1+1+1)^n=\sum_{i+j+k=n}\frac{n!}{i!j!k!}1^i1^j1^k$ .

Conclusion. $\sum_{i+j+k=n}\frac{n!}{i!j!k!}=3^n$ .