What is the coefficient of `x^2y^3z^5` in `(x+y+z)^(10)` ?

Explanation

To find the coefficient of \( x^2y^3z^5 \) in the expansion of \( (x+y+z)^{10} \), we can use the multinomial theorem. The multinomial theorem states that:

\[ (x_1 + x_2 + ... + x_m)^n = \sum_{k_1 + k_2 + ... + k_m = n} \frac{n!}{k_1! k_2! ... k_m!} x_1^{k_1} x_2^{k_2} ... x_m^{k_m} \]

In our case, we have \( x_1 = x \), \( x_2 = y \), \( x_3 = z \), and \( n = 10 \). We want to find the coefficient of \( x^2y^3z^5 \), which means we need \( k_1 = 2 \), \( k_2 = 3 \), and \( k_3 = 5 \).

...