Suppose that we wish to expand (x + y + z)17. what is the coefficient of x 2 y5z10

We use the trinomial theorem to answer this question. Suppose we have a trinomial (a + b + c)ⁿ, we can determine any term to be:

[n!/(n-m)!(m-k)!k!] a^(n-m) b^(m-k) c^k

In this problem, the variables are: x=a, y=b and z=c. We already know the exponents of the variables. So, we equate this with the form of the trinomial theorem.

n - m = 2
m - k = 5
k = 10

Since we know k, we can determine m. Once we know m, we can determine n. Then, we can finally solve for the coefficient.

m - 10 = 5
m = 15

n - 15 = 2
n = 17

Therefore, the coefficient is equal to:

Coefficient = n!/(n-m)!(m-k)!k! = 17!/(17-5)!(15-10)!10! = 408,408