Question

Find r if : (a) ${}^{14}{C}_{2r}:{}^{10}{C}_{2r-4}=143:90$

Answer 1

${}^{14}{C}_{2r}{:}^{10}{C}_{2r-4}\frac{14!}{2r!14-2r!}:\frac{10!}{14-2r!(2r-4)!}\frac{14\times 13\times 12\times 11}{2r!2r-1!2r-2!2r-3!}=\frac{143}{90}$

$2r!(2r-1)!(2r-2)!(2r-3)!=14\times 12\times 9\times 10=7\times 8\times 9\times 102r=10r=5$