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The value of .^(40)C(0) xx .^(100)C(40) ...

The value of `.^(40)C_(0) xx .^(100)C_(40) - .^(40)C_(1) xx .^(99)C_(40) + .^(40)C_(2) xx .^(98)C_(40) "……." + .^(40)C_(40) xx .^(60)C_(40)` is equal to `"____"`.

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The correct Answer is:
1
Given sum
`=` Coefficient of `x^(40)` in `.^(40)C_(0)(1+x)^(100) - .^(40)C_(1)(1+x)^(99) + .^(40)C_(2)(1+x)^(96)+"….."+.^(40)C_(40)(1+x)^(60)`
= Coefficient of `x^(40)` in `(1+x)^(60)(.^(40)C_(0)(1+x)^(40)-.^(40)C_(1)(1+x)^(39)+.^(40)C_(2)(1+x)^(38)+"....."+.^(40)C_(40))`
`=` Coefficient of `x^(40)` in `(1+x)^(60)(1+x-1)^(40)`
= Coefficient of `x^(40)` in `(1+x)^(60)x^(40)`
`= 1`
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