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Question

Assertion :nr=0r(nCr)2=n(2n1Cn1) Reason: nr=0(nCr)2=2nCn

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
Both Assertion and Reason are incorrect
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Solution

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
nr=0r(nCr)2=(nC1)2+2×(nC2)2+3(nC3)2+....+n(nCn)2
Now, (1+x)n=nC0+x(nC1)+x2(nC2)+x3(nC3)+...+xn(nCn)
Taking derivative on both sides,
n(1+x)n1=nC1+2x(nC2)+3x2(nC3)+...+nxn1(nCn)
Also, (x+1)n=nC0xn+nC1xn1+....+nCnx0
Now, nr=0(nCr)2= coefficient of xn in (1+x)2n= 2nCn
Similarly, nr=0r(nCr)2= coefficient of xn1 in n(1+x)n1×(1+x)n=n×(1+x)2n1= n×2n1Cn1

hence, assertion and reason are both correct but reason is not the correct explanation for assertion.

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