Question
Mathematics Question on Combinations
The value of 12.C1+32.C3+52.C5+... is :
A
n(n−1)n−2+n.2n−1
B
n(n−1)n−2
C
n(n−1)n−3
D
none of the above
Answer
none of the above
Explanation
Solution
We know ∑r=1nr2.nCr=n(n−1)2n−2 +n.2n−1 .....(1) and ∑r=1n(−1)r−1.r2.nCr=0 ...(2) Adding (1) & (2) we get 2[12.C1+32.C3+52C5+....] =n(n−1)2n−2+n.2n−1 ⇒[12C1+32C3+52C5+....] =n(n−1)2n−3+n.2n−2.