Let the eleven letters A, B, ....., K denote | vedclass

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

Question

(C) Let $P = (A - 1)(B - 2)(C - 3) ..... (K - 11)$.Consider the sum of the factors: $S = (A - 1) + (B - 2) + (C - 3) + ..... + (K - 11)$.$S = (A + B +

Vedclass · Accepted Answer

Always even

Vedclass · Answer

(C) Let $P = (A - 1)(B - 2)(C - 3) ..... (K - 11)$.Consider the sum of the factors: $S = (A - 1) + (B - 2) + (C - 3) + ..... + (K - 11)$.$S = (A + B + C + ..... + K) - (1 + 2 + 3 + ..... + 11)$.Since $(A, B, ....., K)$ is a permutation of $(1, 2, ....., 11)$,the sum $(A + B + C + ..... + K) = (1 + 2 + 3 + ..... + 11) = 55 + 1 = 66$.Thus,$S = 66 - 66 = 0$.In any set of integers,the sum of the differences $(A_i - i)$ is zero. For the sum of $11$ integers to be $0$ (an even number),there must be an even number of odd terms in the sum.Since there are $11$ terms in total (an odd number),at least one term $(A_i - i)$ must be even.If at least one factor in a product of integers is even,the entire product must be even.Therefore,the product is always even.

Let the eleven letters $A, B, ....., K$ denote an arbitrary permutation of the integers $(1, 2, ....., 11)$. Then,what is the nature of the product $(A - 1)(B - 2)(C - 3) ..... (K - 11)$?

Similar Questions

If $\frac{1}{6!} + \frac{1}{7!} = \frac{x}{8!}$,find $x$.

The total number of positive integral solutions $(x, y, z)$ of $xyz = 24$ is

All the letters of the word $TABLE$ are permuted and the strings of letters (may or may not have meaning) thus formed are arranged in dictionary order. Then the rank of the word $TABLE$ counted from the rank of the word $BLATE$ is

Two packs of $52$ cards are shuffled together. The number of ways in which a man can be dealt $26$ cards so that he does not get two cards of the same suit and same denomination is

The number of ways in which $3$ children can distribute $10$ tickets out of $15$ consecutively numbered tickets among themselves such that they get consecutive blocks of $5, 3$ and $2$ tickets is

Vedclass Test Series

Exam Paper Generator

Online Exam Module