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!)`.

Answer 1

`P_(n+1)/P_n=((n+1)C_0*(n+1)C_1*...(n+1)C_(n+1))/(nC_0*nC_1*...nC_n`
`P_n=n/(1!)*(n(n-1))/(2!)*...(n!)/(n!)`
`=(n^n(n-1)^(n-1)(n-1)^(n-2)...(33)^1)/(1!*2!*3!...n!`
`P(n+1)=(n+1)/1*(n(n+1))/(2!)*(n(n+1)(n-1))/(3!)...(n+1)/(n+1)`
`=((n+1)^(n+1)*(n)^n*(n-1)^(n-1)...1)/(1!*2!*3!...(n+1)!)`
`P(n+1)/P_n=((n+1)^(n+1)/((n+1)!))`
`=(n+1)^n/(n!)`.