Question

The ratio of the coefficients of x^p and x^q in the expansion of (1 + x)^{p + q} is ______.

Answer 1

The ratio of the coefficients of x^p and x^q in the expansion of (1 + x)^{p + q} is 1 : 1.

Explanation:

Given expansion is (1 + x)^{p + q} 

T_r+1 = `""^(p + q)"C"_r  x^r`

Put r = p = `""^(p + q)"C"_p x^p`

∴ The coefficient of x^p = `""^(p + q)"C"_p`

Similarly, coefficient of x^q = `""^(p + q)"C"_q`

`""^(p + q)"C"_p = ((p + q)!)/(p!(p + q - p)!)`

= `((p + q)!)/(p!q!)`

And `""^(p + q)"C"_q = ((p + q)!)/(q!(p + q - q)!)`

= `((p + q)!)/(p!q!)`

So, the ratio is 1 : 1.