sir this is the doubt : sigma r=0 to n(r+1)(nCr)^2. = ?some more are there. sigma r=1 to n(nCr)^2=?.
Manisha , 3 Years ago
Grade 12
Algebra> sir this is the doubt : sigma r=0 to n (r...
1 AnswersYash Patari
Last Activity: 3 Years ago
dear student,
∑r=0n(r+1)Cr2
It can be written as
∑r=0nr⋅Cr2+∑r=0nCr2
We know that
r⋅nCr=n⋅n−1Cr−1
Hence
r⋅Cr2=n⋅n−1Cr−12
∑r=0nn⋅n−1Cr−12=n[n−1C02+n−1C12+.......+n−1Cn−12]=n⋅(n−1)!(2n−1−n+1)!(2n−1)!=n⋅(n−1)!n!(2n−1)!−−−−(1)
∑r=0nCr2=n!n!2n!−−−−−(2)
Adding both eq(1) and (2)
∑r=0n(r+1)Cr2=n⋅(n−1)!n!(2n−1)!+n!n!2n!
Adding both eq(1) and (2)
∑r=0n(r+1)Cr2=n⋅(n−1)!n!(2n−1)!+n!n!2n!
∑r=0n(r+1)Cr2=n⋅(n−1)!n!(2n−1)!+n(n−1)!n!2n(2n−1)!
∑r=0n(r+1)Cr2=(n−1)!n!(n+2)(2n−1)!
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