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If P(n) denotes the product of all the c...

If `P_(n)` denotes the product of all the coefficients of `(1+x)^(n)` and `9!P_(n+1)=10^(9)P_(n)` then `n` is equal to

A

`10`

B

`9`

C

`19`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
`(b)` `P_(n)=^(n)C_(0)^(n)C_(1)^(n)C_(2)....^(n)C_(n)`
`(P_(n+1))/(P_(n))=("^(n+1)C_(1))/('^(n)C_(1))*('^(n+1)C_(2))/('^(n)C_(2))....('^(n+1)C_(n))/('^(n)C_(n)).'^(n+1)C_(n+1)`
`=(n+1)/(n)*(n+1)/(n-1)*(n+1)/(n-2)...(n+1)/(1)=((n+1)^(n))/(n!)`
`implies((n+1)^(n))/(n!)=(10^(9))/(9!)`
`impliesn=9`
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