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Question

Mathematics Question on Combinations

The value of 12.C1+32.C3+52.C5+...1^2.C_1 + 3^2.C_3 + 5^2.C_5 + ... is :

A

n(n1)n2+n.2n1n (n - 1)^{n -2} + n . 2^{n - 1 }

B

n(n1)n2n (n - 1 )^{n - 2 }

C

n(n1)n3n (n - 1 )^{n - 3 }

D

none of the above

Answer

none of the above

Explanation

Solution

We know r=1nr2.nCr=n(n1)2n2\sum^{n}_{r=1} r^{2} .^{n}C_{r} = n\left(n-1\right)2^{n-2} +n.2n1+ n. 2^{n-1} .....(1) and r=1n(1)r1.r2.nCr=0\sum^{n}_{r=1} \left(-1\right)^{r-1} .r^{2} . ^{n}C_{r} = 0 ...(2) Adding (1) & (2) we get 2[12.C1+32.C3+52C5+....]2[1^2 . C_1 + 3^2 . C_3 + 5^2 \, C_5 + ....] =n(n1)2n2+n.2n1= n(n - 1)2^{n-2} + n . 2^{n-1} [12C1+32C3+52C5+....]\Rightarrow \, [1^2 \, C_1 + 3^2 \, C_3 + 5^2 \, C_5 +....] =n(n1)2n3+n.2n2.= n(n - 1)2^{n-3 } + n . 2^{n-2}.