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Find the sum sum(k=0)^(10).^(20)Ck....

Find the sum `sum_(k=0)^(10).^(20)C_k`.

Text Solution

Verified by Experts

`S = underset(k=0)overset(10)sum.^(20)C_(k)`
` = .^(20)C_(0) + .^(20)C_(1) + "……" + .^(20)C_(10)`
Now,
`.^(20)C_(0) + .^(20)C_(1) + "……" + .^(20)C_(9) + .^(20)C_(19) + .^(20)C_(11) + "……" + .^(20)C_(20) = 2^(20)`
or `(.^(20)C_(0) + .^(20)C_(20)) + (.^(20)C_(1) + .^(20)C_(19))+ "....."`
`+ (.^(20)C_(9) + .^(20)C_(11)) + .^(20)C_(10) = 2^(20)`
or `2[.^(20)C_(0) + .^(20)C_(1) + "......." + .^(20)C_(10)] - .^(20)C_(10) = 2^(20)`
`( :' .^(n)C_(r) = .^(n) C_(n-r))`
`:. S = .^(20)C_(0) + .^(20)C_(1) + "......." + .^(20)C_(10)`
`= 2^(19) + 1/2 .^(20)C_(10)`
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