1.

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

Answer»

`2^(10)- 1`
`2^(10)`
`3^(10) - 1`
`3^(10)`

Solution :`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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