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Let X=(\ ^(10)C1)^2+2(\ ^(10)C2)^2+3(\ ^...

Let `X=(\ ^(10)C_1)^2+2(\ ^(10)C_2)^2+3(\ ^(10)C_3)^2+\ ddot\ +10(\ ^(10)C_(10))^2` , where `\ ^(10)C_r` , `r in {1,\ 2,\ ddot,\ 10}` denote binomial coefficients. Then, the value of `1/(1430)\ X` is _________.

Text Solution

Verified by Experts

The correct Answer is:
D
`X = underset(r=1)overset(10)sumr.(.^(10)C_(r))^(2)= underset(r=1)overset(10)sumr..^(10)C_(r)..^(10)C_(r)`
`= 10. underset(r=1)overset(10)sum .^(9)C_(r-1)..^(10)C_(10-r) = 10..^(19)C_(9)`
Now, `(X)/(1430) = (10..^(19)C_(9))/(1430) = (.^(19)C_(9))/(143) = (.^(19)C_(9))/(11xx13)`
`= (19xx17xx16)/(8) = 19xx34 = 646`
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