Question

The coefficient of xkyn-k in the expansion of (x+y)n equals (n/k) true or false

Answer 1

The correct option is 1. The given statement is true.

Explanation:

The binomial expansion is defined as

`(x+y)^n=^nC_0x^(n-0)y^(1)+^nC_(1)x^(n-1)y^(2)+....+^nC_(n-1)x^(1)y^(n-1)+^nC_nx^(0)y^(n)`

The rth term in a binomial expansion is defined as

`\text{rth term}=^nC_rx^(n-r)y^(r)`

Let the coefficient of
`x^ky^(n-k)` be A. The power of x is k and the power of y is n-k. It means

`k=n-r`

`n-k=r`

The coefficient of
`x^ky^(n-k)` is

`^nC_(n-k)=\binom{n}{n-k}`

Using the property of combination,

`^nC_(n-r)=^nC_r`

`^nC_(n-k)=^nC_(k)=\binom{n}{k}`

The coefficient of
`x^ky^(n-k)` is
`\binom{n}{k}`. Therefore the given statement is true.