Let s_1 = sum_{j=1}^{10} j(j-1) binom{10}{j}, | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) We know that $\sum_{j=0}^{n} \binom{n}{j} = 2^n$.For $s_1 = \sum_{j=2}^{10} j(j-1) \binom{10}{j} = \sum_{j=2}^{10} 10 	imes 9 \binom{8}{j-2} = 90

Vedclass · Accepted Answer

Statement $-1$ is true,Statement $-2$ is false

Vedclass · Answer

(D) We know that $\sum_{j=0}^{n} \binom{n}{j} = 2^n$.For $s_1 = \sum_{j=2}^{10} j(j-1) \binom{10}{j} = \sum_{j=2}^{10} 10 	imes 9 \binom{8}{j-2} = 90 	imes 2^8$.For $s_2 = \sum_{j=1}^{10} j \binom{10}{j} = \sum_{j=1}^{10} 10 \binom{9}{j-1} = 10 	imes 2^9$.For $s_3 = \sum_{j=1}^{10} j^2 \binom{10}{j} = \sum_{j=1}^{10} (j(j-1) + j) \binom{10}{j} = s_1 + s_2$.$s_3 = 90 	imes 2^8 + 10 	imes 2^9 = 45 	imes 2^9 + 10 	imes 2^9 = 55 	imes 2^9$.Comparing with the statements:Statement $-1$ is $55 	imes 2^9$,which is true.Statement $-2$ claims $s_2 = 10 	imes 2^8$,but we found $s_2 = 10 	imes 2^9$,so Statement $-2$ is false.Thus,Statement $-1$ is true and Statement $-2$ is false.

Let $s_1 = \sum_{j=1}^{10} j(j-1) \binom{10}{j}$,$s_2 = \sum_{j=1}^{10} j \binom{10}{j}$,and $s_3 = \sum_{j=1}^{10} j^2 \binom{10}{j}$.
Statement $-1$: $s_3 = 55 \times 2^9$
Statement $-2$: $s_1 = 90 \times 2^8$ and $s_2 = 10 \times 2^8$

Similar Questions

The greatest term in the expansion of $\sqrt{3} \left( 1 + \frac{1}{\sqrt{3}} \right)^{20}$ is

The middle term in the expansion of $(1 + x)^{2n}$ is

If the coefficients of the $(2r+1)$-th term and the $(r+2)$-th term in the expansion of $(1+x)^{43}$ are equal,then $r$ is equal to:

If the $2^{\text{nd}}$,$3^{\text{rd}}$,and $4^{\text{th}}$ terms in the expansion of $(x+a)^{n}$ are $96, 216$,and $216$ respectively,and $n$ is a positive integer,then $a+x=$

If the coefficient of $x^9$ in $(\alpha x^3 + \frac{1}{\beta x})^{11}$ and the coefficient of $x^{-9}$ in $(\alpha x - \frac{1}{\beta x^3})^{11}$ are equal,then $(\alpha \beta)^2$ is equal to $.............$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Let $s_1 = \sum_{j=1}^{10} j(j-1) \binom{10}{j}$,$s_2 = \sum_{j=1}^{10} j \binom{10}{j}$,and $s_3 = \sum_{j=1}^{10} j^2 \binom{10}{j}$.Statement $-1$: $s_3 = 55 \times 2^9$Statement $-2$: $s_1 = 90 \times 2^8$ and $s_2 = 10 \times 2^8$