Question

The value of $ \displaystyle\sum_{r=1}^{10} r(r-1) \cdot{ }^{10} C_{r}$ is

• A. $45 \times 2^{10}$
• B. $75 \times 2^{9}$
• C. $2^{9}$
• D. $45 \times 2^{9}$

Answer 1

Answer: (D)

$ \displaystyle\sum_{r=1}^{10} r(r-1)^{10} C_{r}$
$= \displaystyle\sum_{r=1}^{10} r(r-1) \frac{10 !}{r(r-1)(r-2) !(10-r) !}$
$=90 \displaystyle\sum_{r=2}^{10}{ }^{8} C_{r-2}$
$=90 \times 2^{8}=45 \times 2^{9}$