JEE Main 2021 · 24 Feb · Shift 1 · Mathematics Q64

MathematicsSingle correct+4 / -1Question 64 of 90

The value of $-{}^{15}C_1+2\cdot{}^{15}C_2-3\cdot{}^{15}C_3+\dots-15\cdot{}^{15}C_{15}+{}^{14}C_1+{}^{14}C_3+{}^{14}C_5+\dots+{}^{14}C_{11}$ is :

2^{16}-1

2^{13}-14

2^{13}-13

2^{14}

Answer and worked solution

Official answer: B

First part. $\sum_{r=1}^{15}(-1)^r r\binom{15}r=-15\sum(-1)^{r-1}\binom{14}{r-1}=-15(1-1)^{14}=0$ .

Second part. Odd binomials of 14 sum to $2^{13}$ ; dropping ${}^{14}C_{13}=14$ leaves $2^{13}-14$ .

(B)

Topics:Properties of binomial coefficients

JEE Main 2021 · 24 Feb · Shift 1, Mathematics question 64