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The value of sum(r=1)^n(sum(p=0)^(r-1) ^...

The value of `sum_(r=1)^n(sum_(p=0)^(r-1) ^nC_r ^rC_p 2^p)` is equal to (a) `4^(n)-3^(n)+1` (b) `4^(n)-3^(n)-1` (c) `4^(n)-3^(n)+2` (d) `4^(n)-3^(n)`

A

`4^(n)-3^(n)+1`

B

`4^(n)-3^(n)-1`

C

`4^(n)-3^(n)+2`

D

`4^(n)-3^(n)`

Text Solution

Verified by Experts

The correct Answer is:
D
`(d)` `sum_(r=1)^(n)"^(n)C_(r )((1+2)^(r )-2^(r ))`
`=sum_(r=1)^(n)"^(n)C_(r )*3^(r )-sum_(r=1)^(n)'^(n)C_(r )*2^(r )`
`=(4^(n)1-1)-(3^(n)-1)=4^(n)-3^(n)`
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