Multinomial coefficient question.

[MathematicalPhysicist]

let r>1 which term in (x1+...+xk)^rk has the greatest coefficient?
well i have this equation:
$$(x_1+x_2+...+x_k)^{rk}=\sum_{n_1+n_2+...+n_k=rk}\left(\begin{array}{cc}rk\\\ n_1,n_2,...,n_k\end{array}\right)x^{n_1}...x^{n_k}$$
well if we notice that (n_1+...+n_k)/k=r then the maximum coefficient is achieved when n_1=n_2=...=n_k=r, but the only way i can see how show that this is true is with lagrange multipliers, and i haven't yet used this method in my calclulus classes so i guess there's a combinatorial solution here. anyone care to hint me this method?

thanks in advance.

 

[MathematicalPhysicist]

well i think i solved it.
if one of n_k's is smaller than r then there must be another one that is bigger than r and so we will have the coeffiecient smaller than the one achieved by n1=...=nk=r.
this is why we get that this must be hthe maximum coefficient.

 

[tacman]

The term with the greatest coefficient in (x1+...+xk)^rk is the one where all the exponents are equal to r. This can be seen by using the multinomial formula, which shows that the coefficient is equal to the multinomial coefficient (rk choose n1, n2,...,nk). Since the sum of the exponents must equal rk, the only way to maximize this coefficient is to have all the exponents equal to r. This can also be seen intuitively, as having all the exponents equal to r would result in the largest number of terms being multiplied together, thus giving the greatest coefficient. This can be proven using the method of Lagrange multipliers, but there is also a combinatorial solution as shown above.