Question

If ²⁸C_2r : ²⁴C_{2r − 4} = 225 : 11, find r.

Answer 1

We have, ²⁸C_2r : ²⁴C_{2r − 4} = 225 : 11
$$\begin{gathered} \Rightarrow \frac{{}^{28}{C}_{2r}}{{}^{24}{C}_{2r-4}}=\frac{225}{11} \\ \Rightarrow \frac{28!}{2r!(28-2r)!}\times \frac{(2r-4)!(28-2r)!}{24!}=\frac{225}{11} \\ \Rightarrow \frac{28\times 27\times 26\times 25}{2r(2r-1)(2r-2)(2r-3)}=\frac{225}{11} \\ \Rightarrow 2r(2r-1)(2r-2)(2r-3)=\frac{28\times 27\times 26\times 25\times 11}{225} \\ \Rightarrow 2r(2r-1)(2r-2)(2r-3)=28\times 3\times 26\times 11 \\ \Rightarrow 2r(2r-1)(2r-2)(2r-3)=4\times 7\times 3\times 13\times 2\times 11 \\ \Rightarrow 2r(2r-1)(2r-2)(2r-3)=(2\times 7)\times 13\times (3\times 4)\times 11 \\ \Rightarrow 2r(2r-1)(2r-2)(2r-3)=14\times 13\times 12\times 11 \\ \Rightarrow 2r=14 \\ \Rightarrow r=7 \end{gathered}$$