If some three consecutive coefficients in the binomial expansion of (x+1)n in powers of x are in the ratio 2:15:70, then the average of these three coefficients is:
Solutions · 1
Official solution
nCr−1:nCr:nCr+1=2:15:70
\[\frac{{}^nC_{r-1}}{{}^nC_r}=\frac{r}{n-r+1}=\frac{2}{15}\quad\Rightarrow\quad17r=2n+2\quad\text{...(i)}\]
\[\frac{{}^nC_r}{{}^nC_{r+1}}=\frac{r+1}{n-r}=\frac{15}{70}=\frac{3}{14}\quad\Rightarrow\quad17r=3n-14\quad\text{...(ii)}\]
From (i),(ii): 2n+2=3n−14⇒n=16,r=2.
\[\text{Average}=\frac{{}^{16}C_1+{}^{16}C_2+{}^{16}C_3}{3}=\frac{16+120+560}{3}=\frac{696}{3}=232\]