I was studying the multinomial theorem: $(u_1+u_1+...u_k)^n=\sum\limits_{r_1+r_2+...r_k=n}\dfrac{n!}{r_1!r_2!...r_k!}u_1^{r_1}u_2^{r_2}...u_k^{r_k}$

and my book said that the number of terms in the expansion of a polynomial using the multinomial theorem is equal to the the number of solutions of

the equation $r_1 + r_2 + .... + r_k = n$

$n \in N$;

$0\leq r_i \leq n $; $r_i \in W$; $i=0,1,2,..,k$;

I know that the answer to this is $\binom{n+k-1}{k-1}$ but I do not know how to get this result. Please help me out here.


It can be put like this
You want to put n like things in k baskets
So there will be $ k-1$ barriers and the permutations of n like objects and $k-1$ barriers are
$$ \frac{^{n+k-1}P_{k-1}}{n!} $$ Which is $$ ^{n+k-1}C_{k-1} $$

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.