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The value of .^(20)C(0)+.^(20)C(1)+.^(20...

The value of `.^(20)C_(0)+.^(20)C_(1)+.^(20)C_(2)+.^(20)C_(3)+.^(20)C_(4)+.^(20)C_(12)+.^(20)C_(13)+.^(20)C_(14)+.^(20)C_(15)` is

A

A. `2^(19)-((.^(20)C_(10)+.^(20)C_(9)))/(2)`

B

B. `2^(19)-((.^(20)C_(10)+2xx.^(20)C_(9)))/(2)`

C

C. `2^(19)-(.^(20)C_(10))/(2)`

D

D. none of these

Text Solution

Verified by Experts

The correct Answer is:
B
Given series is `.^(20)C_(0) + .^(20)C_(1) + .^(20)C_(2) + "….." + .^(20)C_(5)`
`= 1/2(2..^(20)C_(0) + 2.^(20)C_(1) + "….." 2..^(20)C_(8))`
`= 1/2[(.^(20)C_(0) + .^(20)C_(1) + "….." +.^(20)C_(8) + .^(20)C_(9) +.^(20)C_(10) + .^(20)C_(11) + "….." + .^(20)C_(20))-(.^(20)C_(9) + .^(20)C_(10) + .^(20)C_(11))]`
`= 1/2[2^(20) - 2..^(20)C_(9)-.^(20)C_(10)]`
`= 2^(19)- ((2..^(20)C_(9) + .^(20)C_(10)))/(2)`
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