Home
Class 11
MATHS
If n be a positive integer and Pn denote...

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!)`.

Text Solution

Verified by Experts

`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!)`.
Promotional Banner

Similar Questions

Explore conceptually related problems

If n be a positive interger 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!) .

If P_(n) denotes the product of all the coefficients in the expansion of (1+x)^(n), n in N , show that, (P_(n+1))/(P_(n))=((n+1)^(n))/(n!) .

If n is a positive integer then .^(n)P_(n) =

If n is a positive integer then the coefficient of x ^(-1) in the expansion of (1+x) ^(n) (1+ (1)/(x)) ^(n) is-

If n is a positive integer and r is a nonnegative integer, prove that the coefficients of x^r and x^(n-r) in the expansion of (1+x)^(n) are equal.

If P_n denotes the product of all the coefficients of (1+ x)^n and 8! P_(n+1)=9^8 P_n then n is equal to

If m and n are positive integers, then prove that the coefficients of x^(m) " and " x^(n) are equal in the expansion of (1+x)^(m+n)

If m and n are positive integers, then prove that the coefficients of x^(m) " and " x^(n) are equal in the expansion of (1+x)^(m+n)