Question

Calculate the coefficient of x4y18 in the expansion of (x − y)22

Answer 1

Binomial theorem:


`(x-y)^n=\displaystyle\sum_(k=0)^n\binom nkx^k(-y)^(n-k)`

The term with
`x^4y^(18)` corresponds to
`n=18+4=22` and
`k=4` (or
`k=18`, since
`\dbinom nk=\dbinom n{n-k}`).

So, the coefficient of this term is


`\dbinom{22}4(-1)^(18)=\dbinom{22}4=(22!)/(4!(22-4)!)=7315`