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Question

If n is a positive integer, prove that nr=1r3(nCrnCr1)2=n(n+1)2(n+2)12.

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Solution

nCr=n!(nr)!.r!
=(n(r1))n!(n(r1))(nr)!.(r1)!.r
=(n(r1))n!(n(r1)!)(r1)!.r
=n(r1)rnCr1
Hence, nCrnCr1=n+1rr
Hence, (nCrnCr1)2=(n+1r)2r2
r3(nCrnCr1)2=r(n+1r)2
=r[(n+1)22r(n+1)+r2]
=r(n+1)22r2(n+1)+r3

r3(nCrnCr1)2=r(n+1)22r2(n+1)+r3
=(n+1)2r2(n+1)r2+r3
=(n+1)2(n(n+1)2)(2(n+1))(n(n+1)(2n+1)6)+n2(n+1)24
=n(n+1)322n(n+1)2(2n+1)6+n2(n+1)24
=n(n+1)22[(n+1)2(2n+1)3+n2]
=n(n+1)22[6n+68n4+3n6]
=n(n+1)22[n+26]
=n(n+1)2(n+2)12

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