Question

The sum of values of $r$ for which ${}^{18}{C}_{r-2}+2\cdot {}^{18}{C}_{r-1}+{}^{18}{C}_r\ge {}^{20}{C}_{13}$

• A. $40$
• B. $50$
• C. $60$
• D. $70$

Answer 1

Answer: (D)

Using formula ${}^n{C}_r+{}^n{C}_{r-1}={}^{n+1}{C}_r$
${}^{18}{C}_{r-2}+{}^{18}{C}_{r-1}+{}^{18}{C}_{r-1}+{}^{18}{C}_r\ge {}^{20}{C}_{13}$
${}^{19}{C}_{r-1}+{}^{19}{C}_r\ge {}^{20}{C}_{13}$
${}^{20}{C}_r\ge {}^{20}{C}_{13}\text{ or }{}^{20}{C}_{20-r}\ge {}^{20}{C}_{13}$
$r=7,8,9,10,11,12,13$
Sum of all values of $r=70$.