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Question

# Assertion :For n∈N, let Cr=(nr), 0≤r≤n∑nr=1(r)C2r=n(2n−1n−1) Reason: ∑nr=1C2r=(2nn)

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Assertion is incorrect but Reason is correct
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Solution

## The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for AssertionFor truth of reasonWe have C20+C21+C22+...+C2n=n∑r=0(nCr)(nCr)=n∑r=0(nCr)(nCn−r)[∵nCr=nCn−r]= number of ways of choosing n persons out of n men and n women= number of ways of choosing n persons out of 2n persons=2nCn For r≥1,r Cr=rn!r!(n−r)!=n(n−1r−1)∴n∑r=1rCr=nn∑r=1(n−1r−1)=n(n+n−1n−1)=n(2n−1n−1)Hence, both reason and assertion are correct.

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