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The value of .^(40)C0+.^40C1 + .^40C2 + ...

The value of `.^(40)C_0+.^40C_1 + .^40C_2 + + .^40C_20`

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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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