If we wish to expand (x + y) 8, what is the coefficient of x 5 y 3? what is the coefficient of x 3 y 5?

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

[n! / (n-r) ! r!] a^ (r) b^ (n-r)

a.) For x⁵y³, the variables are: x=a and y=b. We already know the exponents of the variables. So, we equate this with the form of the binomial theorem.

r = 5

n - r = 3

Solving for n,

n = 3 + 5 = 8

Therefore, the coefficient is equal to:

Coefficient = n! / (n-r) ! r! = 8! / (8-5) !8! = 56

b.) For x³y⁵, the variables are: x=a and y=b. We already know the exponents of the variables. So, we equate this with the form of the binomial theorem.

r = 3

n - r = 5

Solving for n,

n = 5 + 3 = 8

Therefore, the coefficient is equal to:

Coefficient = n! / (n-r) ! r! = 8! / (8-3) !8! = 56