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Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Arbitrary Arrangement.

## Arbitrary Arrangement ( B.Stat Objective Question )

Let $a_1, a_2, ….,a_{11}$ be an arbitrary arrangement (ie permutation) of the integers 1,2,….,11. Then the numbers $(a_1-1)(a_2-2)….(a_{11}-{11})$ is

• necessarily $\leq$ 0
• necessarily even
• necessarily 0
• none of these

### Key Concepts

Permutation

Numbers

Even and Odd

But try the problem first…

Source

B.Stat Objective Problem 119

Challenges and Thrills of Pre-College Mathematics by University Press

## Try with Hints

First hint

necessarily $\leq$ 0 case

taking values (2-1)(3-2)(4-3)…..(10-9)(1-10)(10-11)

here all the terms except last two terms are positive and there are 2 negetive terms whose product will be even

then product > 0

then not necessarily < 0 or = 0

Second Hint

necessarily even case

we assume that the product is not necessarily even

that is each of the factor have to be odd

then the arrangement look like

(even-1)(odd-2)(even-3)(odd-4)….(even-9)(odd-10)

but only one odd number left which will pair with 11 that a contradiction

Final Step

$\Rightarrow$ product is even

$\Rightarrow$ necessarily even.