Question

The value of $(\frac{50}{0})(\frac{50}{1})+(\frac{50}{1})(\frac{50}{2})+....+(\frac{50}{49})(\frac{50}{50})$ is, where ${}^n{C}_r=(\frac{n}{r})$

• A. $(\frac{100}{50})$
• B. $(\frac{100}{51})$
• C. $(\frac{50}{25})$
• D. ${(\frac{50}{25})}^2$

Answer 1

Answer: (B)

$(\frac{100}{51})$

Given

$(\frac{50}{0})(\frac{50}{1})+(\frac{50}{1})(\frac{50}{2})+.......+(\frac{50}{49})(\frac{50}{50})$

We know that

${}^n{C}_0{}^n{C}_1+{}^n{C}_1{}^n{C}_2+..........+{}^n{C}_{n-1}{}^n{C}_n={}^{2n}{C}_{n-1}$

Hence,

$(\frac{50}{0})(\frac{50}{1})+(\frac{50}{1})(\frac{50}{2})+.......+(\frac{50}{49})(\frac{50}{50})=(\frac{100}{49})=(\frac{100}{51})$