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

To address this question, we apply the trinomial theorem. Given a trinomial (a + b + c)ⁿ, we can identify any term using:

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

For this scenario, x=a, y=b, and z=c are the variables provided. We already know the exponents associated with each variable. Thus, we set this equal to the trinomial theorem's structure.

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

With k known, we can now find m. Once m is determined, n can be found as well. Finally, we solve for the coefficient.

m - 10 = 5
m = 15

n - 15 = 2
n = 17

Thus, the coefficient is calculated to be:

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