InterviewSolution
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InterviewSolution
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Statement -1: ` sum_(r=0)^(n) r(""^(n)C_(r))^(2) = n (""^(2n -1)C_(n-1))` Statement-2: ` sum_(r=0)^(n) (""^(n)C_(r))^(2)= ""^(2n)C_(n)`A. 1B. 2C. 3D. 4 |
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Answer» Correct Answer - b For truth of statement-2, see illustration 17 on page 14,16. Now, `sum_(r=0)^(n) rC_(r)""^(2)= sum_(r=0)^(n) r .""^(n)C_(r).""^(n)C_(r).=` `rArr sum_(r=0)^(n) rC_(r)""^(2)=n sum_(r=0)^(n) ""^(n-1)C_(r-1).""^(n)C_(r).` `rArr sum_(r=0)^(n) rC_(r)""^(2)=n .sum_(r=0)^(n) ""^(n-1)C_(n-r).""^(n)C_(r).` `rArr sum_(r=0)^(n) rC_(r)""^(2)=n` Coefficient of `x^(n)` in `{ (1 + x)^(n -1) (1 + x)^(n)}` `rArr sum_(r=0)^(n) rC_(r)""^(2)=n` Coefficient of `x^(n)` in ` (1 + x)^(2n+1)` `rArr sum_(r=0)^(n) rC_(r)""^(2)=n^(2)""^(n-1)C_(n) = n^(2)""^(n-1)C_(n-1)`. So, statement-1 is also true. But , statement-2 is not a correct expanation for statement -1. |
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