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The sum sumsum(0leilejle10) (""^(10)C(j)...

The sum `sumsum_(0leilejle10) (""^(10)C_(j))(""^(j)C_(i-1))` is equal to

A

A. `2^(10) - 1`

B

B. `2^(10)`

C

C. `3^(10) - 1`

D

D. `3^(10)`

Text Solution

Verified by Experts

The correct Answer is:
C
`underset(0leilejle10)(sum)(.^(10)C_(j))(.^(j)C_(i))`
`= .^(10)C_(1)(.^(1)C_(0) + .^(1)C_(1))+.^(10)C_(2)(.^(2)C_(0)+.^(2)C_(1) + .^(2)C_(2))+"...."+.^(10)C_(10)(.^(10)C_(0) +.^(10)C_(1)+"..."+.^(10)C_(10))`
`= .^(10)C_(1)(2) + .^(10)C_(2)(2^(2))+.^(10)C_(3)(2^(3))+"....."+.^(10)C_(10)(2^(10))`
`= (2+1)^(10) - 1`
`= 3^(10) - 1`
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