Question

If n be a positive integer and `P_n` denotes the product of the binomial coefficients in the expansion of `(1 +x)^n`, prove that `(P_(n+1))/P_n=(n+1)^n/(n!)`.
A. `(n+1)/(n!) `
B. `(n^(n))/(n!)`
C. `((n+1)^(n)) /((n+1)!)`
D. `(n + 1 ^(n+1))/((n +1)!)`

Answer 1

Correct Answer - d
We have ,
`P_(n) = ""^(n)C_(0). ""^(n)C_(1).""^(n)C_(2)...""^(n)C_(n)`
and `P_(n+1) = ""^(n+1)C_(0). ""^(n+1)C_(1).""^(n+1)C_(2)...""^(n+1)C_(n+1)`
`therfore (P_(n+1) )/(P_(n))= (""^(n+1)C_(0). ""^(n+1)C_(1).""^(n+1)C_(2)...""^(n+1)C_(n+1))/(""^(n)C_(0).""^(n)C_(1). ""^(n)C_(2)...""^(n)C_(n))`
`rArr (P_(n+1) )/(P_(n))=( (""^(n+1)C_(1))/(""^(n)C_(0))).(( ""^(n+1)C_(2))/(""^(n)C_(1))).((""^(n+1)C_(3))/(""^(n)C_(2)))...((""^(n+1)C_(n+_1))/( ""^(n)C_(n)))`
`rArr (P_(n+1) )/(P_(n))=( ((n+1)/(1).""^(n)C_(0))/(""^(n)C_(0))).(( (n+2)/(2).""^(n)C_(1))/(""^(n)C_(1))).(((n+3)/(3).""^(n+1)C_(2))/(""^(n)C_(2)))...(((n+1)/(n+1).""^(n)C_(n))/( ""^(n)C_(n)))`
`rArr (P_(n+1) )/(P_(n))=( (n+1)/(1))( (n+1)/(2))( (n+1)/(3))...( (n+1)/(n+1))=( (n+1)^(n+1))/((n+1)!)`