Answer to Question #301937 in Discrete Mathematics for Tege

Question

9. In the expression (1+x⁵+x⁹)¹⁰ find the coefficient of a) x²³ b) x³²

Answer 1

In multinomial expansion of $(1+x^5+x^9)^{10},$ every term will be of form:

$$C(10; m, n, p)\times1^m\times(x^5)^n\times (x^9)^p,$$

where $m + n + p =10.$

a) For the coefficient of $x^{23},$ $(5n + 9p)$ should be equal to $23.$ There is only one pair exist for this condition to hold i.e $(1,2).$

$(m,n,p)$ will be $(7, 1, 2).$

$$C(10; 7, 1, 2)=\dfrac{10!}{7!1!2!}=\dfrac{10(9)(8)}{2}=360$$

The coefficient of $x^{23}$ is $360.$

b) For the coefficient of $x^{32},$ $(5n + 9p)$ should be equal to $32.$ There is only one pair exist for this condition to hold i.e $(1,3).$

$(m,n,p)$ will be $(6, 1, 3).$

$$C(10; 6, 1, 3)=\dfrac{10!}{6!1!3!}=\dfrac{10(9)(8)(7)}{6}=840$$

The coefficient of $x^{32}$ is $840.$