Official solution
Given \[{}^n{C_{r - 1}}:{}^n{C_r}:{}^n{C_{r + 1}} = 2:15:70\]
\[{{{}^n{C_{r - 1}}} \over {{}^n{C_r}}} = {2 \over {15}}\]
\[ \Rightarrow {r \over {n - r + 1}} = {2 \over {15}}\]
\[ \Rightarrow 15r = 2n - 2r + 2\]
\[ \Rightarrow 17r = 2n + 2\] .... (i)
Now \[{{{}^n{C_r}} \over {{}^n{C_{r + 1}}}} = {{15} \over {70}}\]
\[ \Rightarrow {{r + 1} \over {n - r}} = {3 \over {14}}\]
\[ \Rightarrow \] 14r + 14 = 3n -- 3r
\[ \Rightarrow \] 17r = 3n -- 14 ... (ii)
Now From (i) and (ii) equation
2n + 2 = 3n - 14 \[ \Rightarrow \] n = 16
By putting n = 16 in equation (i)
\[ \Rightarrow \] r = 2
\[ \therefore \] Average of coefficient = \[{{{}^{16}{C_1} + {}^{16}{C_2} + {}^{16}{C_3}} \over 3} \]
= \[{{16 + 120 + 560} \over 3}\]
= \[{{696} \over 3} = 232\]