JEE Main 2022 · 30 Jul · Shift 2 · Mathematics Q65

MathematicsSingle correct+4 / -1Question 65 of 90

The sum of all values of n, for which ${}^{n-1}C_4-{}^{n-1}C_3-\dfrac54\,{}^{n-2}P_2<0$ , is

Answer and worked solution

Official answer: B

Simplify with $m=n-1\ge4$ : ${}^mC_4-{}^mC_3=\frac{m(m-1)(m-2)(m-7)}{24}$ and ${}^{n-2}P_2=(m-1)(m-2)$ .

Inequality $\frac{m(m-7)}{24}<\frac54$ : $m^2-7m-30<0$ , so $m<10$ : $m=4,\dots,9$ , $n=5,\dots,10$ . Sum 45.

(B)

Topics:Combinations

JEE Main 2022 · 30 Jul · Shift 2, Mathematics question 65