Try this problem from Test of Mathematics, TOMATO Objective problem number 44, useful for ISI B.Stat and B.Math.
Problem: TOMATO Objective 44
Suppose that $\mathbf{ x_1 , \cdots , x_n}$ (n> 2) are real numbers such that x $\mathbf{x_i = -x_{n-i+1}}$ for $\mathbf{1\le i \le n}$ . Consider the sum $\mathbf{ S = \sum \sum \sum x_i x_j x_k }$ where the summations are taken over all i, j, k: $\mathbf{ 1\le i, j, k \le n }$ and i, j, k are all distinct. Then S equals:
(A) $\mathbf{n!x_1 x_2 \cdots x_m }$ ; (B) (n-3)(n-4); (C) (n-3)(n-4)(n-5); (D) none of the foregoing expressions;
Discussion:
$\mathbf {( x_1 + x_2 + ... + x_n )^3} $
$\mathbf{= \sum \sum \sum {x_i x_j x_k }+ \sum x_i ^2 ( \sum x_j ) + \sum x_i^3} $
Since $\mathbf {x_1 = - x_n} $
Hence $\mathbf {x_1 ^3 = -x_n ^3} $
Since $\mathbf{\sum {x_i} = 0 } $ and $\mathbf{\sum {x_i}^3 = 0} $
Therefore $\mathbf{\sum \sum \sum {x_i x_j x_k } = 0} $.
Hence option D
How to use invariance in Combinatorics – ISI Entrance Problem – Video
Try this problem from Test of Mathematics, TOMATO Objective problem number 44, useful for ISI B.Stat and B.Math.
Problem: TOMATO Objective 44
Suppose that $\mathbf{ x_1 , \cdots , x_n}$ (n> 2) are real numbers such that x $\mathbf{x_i = -x_{n-i+1}}$ for $\mathbf{1\le i \le n}$ . Consider the sum $\mathbf{ S = \sum \sum \sum x_i x_j x_k }$ where the summations are taken over all i, j, k: $\mathbf{ 1\le i, j, k \le n }$ and i, j, k are all distinct. Then S equals:
(A) $\mathbf{n!x_1 x_2 \cdots x_m }$ ; (B) (n-3)(n-4); (C) (n-3)(n-4)(n-5); (D) none of the foregoing expressions;
Discussion:
$\mathbf {( x_1 + x_2 + ... + x_n )^3} $
$\mathbf{= \sum \sum \sum {x_i x_j x_k }+ \sum x_i ^2 ( \sum x_j ) + \sum x_i^3} $
Since $\mathbf {x_1 = - x_n} $
Hence $\mathbf {x_1 ^3 = -x_n ^3} $
Since $\mathbf{\sum {x_i} = 0 } $ and $\mathbf{\sum {x_i}^3 = 0} $
Therefore $\mathbf{\sum \sum \sum {x_i x_j x_k } = 0} $.
Hence option D
How to use invariance in Combinatorics – ISI Entrance Problem – Video
