Question
The value of $${1^2} \cdot {C_1} + {3^2} \cdot {C_3} + {5^2} \cdot {C_5} + .....$$ is :
A.
$$n{\left( {n - 1} \right)^{n - 2}} + n \cdot {2^{n - 1}}$$
B.
$$n{\left( {n - 1} \right)^{n - 2}}$$
C.
$$n{\left( {n - 1} \right)^{n - 3}}$$
D.
None of these
Answer :
None of these
Solution :
We know, $$\sum\limits_{r = 1}^n {{r^2} \cdot {\,^n}{C_r}} = n\left( {n - 1} \right){2^{n - 2}} + n \cdot {2^{n - 1}}\,\,\,\,.....\left( 1 \right)$$
$${\text{and }}\sum\limits_{r = 1}^n {{{\left( { - 1} \right)}^{r - 1}} \cdot {r^2} \cdot {\,^n}{C_r}} = 0\,\,\,\,\,.....\left( 2 \right)$$
Adding (1) & (2) we get,
$$\eqalign{
& 2\left[ {{1^2} \cdot {C_1} + {3^2} \cdot {C_3} + {5^2} \cdot {C_5} + .....} \right] = n\left( {n - 1} \right){2^{n - 2}} + n \cdot {2^{n - 1}} \cr
& \Rightarrow \left[ {{1^2}{C_1} + {3^2}{C_3} + {5^2}{C_5} + .....} \right] = n\left( {n - 1} \right){2^{n - 3}} + n \cdot {2^{n - 2}}. \cr} $$