Question

In the expansion of (1 + x)ⁿ the binomial coefficients of three consecutive terms are respectively 220, 495 and 792, find the value of n.

Answer 1

^{$$\begin{gathered} \mathrm{Suppose}\mathrm{the}\mathrm{three}\mathrm{consecutive}\mathrm{terms}\mathrm{are}{T}_{r-1},T_r\mathrm{and}{T}_{r+1}. \\ \mathrm{Coefficients}\mathrm{of}\mathrm{these}\mathrm{terms}\mathrm{are}{}^{n}C_{r-2},{}^{n}C_{r-1}\mathrm{and}{}^{n}C_r,\mathrm{respectively}. \\ \mathrm{These}\mathrm{coefficients}\mathrm{are}\mathrm{equal}\mathrm{to}220,495\mathrm{and}792. \\ \therefore \frac{{}^{n}C_{r-2}}{{}^{n}C_{r-1}}=\frac{220}{495} \\ \Rightarrow \frac{r-1}{n-r+2}=\frac{4}{9} \\ \Rightarrow 9r-9=4n-4r+8 \\ \Rightarrow 4n+17=13r...\left(1\right) \\ \mathrm{Also}, \\ \frac{{}^{n}C_r}{{}^{n}C_{r-1}}=\frac{792}{495} \\ \Rightarrow \frac{n-r+1}{r}=\frac{8}{5} \\ \Rightarrow 5n-5r+5=8r \\ \Rightarrow 5n+5=13r \\ \Rightarrow 5n+5=4n+17\left[\mathrm{From}\mathrm{Eqn}\left(1\right)\right] \\ \Rightarrow n=12 \end{gathered}$$}