The value of `(.^(21)C_(1) - .^(10)C_(1)) + (.^(21)C_(2) - .^(10)C_(2)) + (.^(21)C_(3) - .^(10)C_(3)) + (.^(21)C_(4) - .^(10)C_(4)) + "…." + (.^(21)C_(10) - .^(10)C_(10))` is

Explanation

`(C(21,1) - C(10,1)) + (C(21,2) - C(10,2)) +...(C(21,10) - C(10,10))`

Now, we know, `C(21,0) = C(10,0) = 1`

So, we can write the given exprssion as ,

`(C(21,0) - C(10,0))+ (C(21,0) - C(10,1)) + (C(21,2) - C(10,2)) +...(C(21,10) - C(10,10))`

`=[C(21,0) + C(21,1) +C(21,2) +...+ C(21,10) ] - [C(10,0)+C(10,1)+C(10,2)+...+C(10,10)]`

`=1/2[C(21,0) + C(21,1) +C(21,2) +...+ C(21,21) ] - [C(10,0)+C(10,1)+C(10,2)+...+C(10,10)]`

`= 1/2(2^21) - 2^10`

`=2^20-2^10.`

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