Correct option is D) 15
\(\displaystyle\sum_{r=0}^{30} r(^{30}C_r)^2\)
\(\displaystyle\sum_{r=0}^{30} r.\,^{30}C_r \times^{30}C_r\)
\(∑ r .\frac{30}{r}. ^{29}C_{r_{-1}} \times ^{30}C_r\)
\(30\displaystyle\sum_{r=1}^{30} \) \(^{29}C_{r_{-1}} \times ^{30}C_r\)
\(30∑ \,^{29}C_{r_{-1}}\times ^{30}C_{30-r}\)
\(30\times\,^{59}C_{29}\)
\(30\times\frac{59!}{29!\times30!}\times\frac{60}{30}\times\frac{30}{60}\)
\(30\times\frac{60!}{(30!)^2}\times\frac{1}{2}\)
\(15\times\frac{60!}{(30!)^2}\)