Question

If $C_0,C_1,C_{2,}\dots ,C_{15}$ 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,
$\frac{{}^n{C}_I}{{}^n{C}_{I-1}}=\frac{n-(r-1)}{r}$
$\Rightarrow r\cdot \frac{{}^n{C}_I}{{}^n{C}_{I-1}}=n+1-r$
$\Rightarrow {\sum }_{I=1}^n\frac{{}^{15}{C}_r}{{}^{15}{C}_{I-1}}={\sum }_{I=1}^{15}(16-r)$
$=16\times 15-\frac{15\times 16}{2}$
$=240-120$
$=120$