Question

If $C_{0},C_{1},C_{2,}\ldots,C_{15}$ are binomial coefficients in (1 +$x$ )¹⁵, then $\frac{C_{1}}{C_{0}}+2\frac{C_{2}}{C_{1}}+3\frac{C_{3}}{C_{2}}+\cdots+15\frac{C_{15}}{C_{14}}=$

• A. 60
• B. 120
• C. 64
• D. 124
• E. 144

Answer 1

Answer: (B)

We know that,
$\frac{{ }^{n} C_{I}}{{ }^{n} C_{I-1}}=\frac{n-(r-1)}{r}$
$\Rightarrow r \cdot \frac{{ }^{n} C_{I}}{{ }^{n} C_{I-1}}=n+1-r $
$\Rightarrow \sum_{I=1}^{n} \frac{{ }^{15} C_{r}}{{ }^{15} C_{I-1}}=\sum_{I=1}^{15}(16-r) $
$= 16 \times 15-\frac{15 \times 16}{2} $
$= 240-120 $
$= 120$