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The sum 2 xx .^(40)C(2) + 6 xx .^(40)C(3...

The sum `2 xx .^(40)C_(2) + 6 xx .^(40)C_(3) + 12 xx .^(40)C_(4) + 20 xx .^(40)C_(5) + "…." + 1560 xx .^(40)C_(40)` is divisible by

A

`3`

B

`5`

C

`13`

D

`2^(41)`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
`2 xx .^(40)C_(2) + 4 xx .^(40)C_(3) + 12 xx .^(40)C_(4) + 20 xx .^(40)C_(5)+ "……." + 1560 xx .^(40)C_(40)`
`= underset(r=2)overset(40)sum(r^(2)-r)..^(40)C_(r)`
`= underset(r=2)overset(40)sumr(r-1)..^(40)C_(r)`
`= underset(r=2)overset(40)sum40(r-1)..^(39)C_(r-1)`
`= underset(r=2)overset(40)sum40 xx 39..^(39)C_(r-2)`
`= 40 xx 39 xx 2^(38)`
`= 40 xx 39 xx 2^(38)`
`= 5 xx 39 xx 2^(41)`
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