The sum of series ^^(20)C0-^^(20)C1+^^(2...

The sum of series `^^(20)C0-^^(20)C1+^^(20)C2-^^(20)C3++^^(20)C 10` is `1/2` `^^(20)C 10` b. `0` c. `^^(20)C 10` d. `-^^(20)C 10`

`""^(20)C_(10)`

`- ""^(20)X_(10)`

`(1)/(2) ""^(20)C_(11)`

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The correct Answer is:

We have,
`""^(20)C_(0)-""^(20)C_(1)+ ""^(20)C_(2)-""^(20)C_(3)+...+ ""^(20)C_(10)-""^(20)C_(11) = 0`
`rArr 2(""^(20)C_(0)-""^(20)C_(1)+ ""^(20)C_(2)-""^(20)C_(3)+...+ ""^(20)C_(9)-""^(20)C_(10) ) - ""^(20)C_(10) = 0`
` [ because ""^(n)C_(r) = ""(n)C_(n-r)]`
`rArr ""^(20)C_(0)-""^(20)C_(1)+ ""^(20)C_(2)-""^(20)C_(3)+...+ ""^(20)C_(9)-""^(20)C_(10) = (1)/(2) ""^(20)C_(10)` .

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