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May 3, 2020

Arbitrary Arrangement | TOMATO B.Stat Objective 119

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

Check the Answer


Answer: necessarily even.

B.Stat Objective Problem 119

Challenges and Thrills of Pre-College Mathematics by University Press

Try with Hints


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

necessarily even case

by contradiction

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

\(\Rightarrow\) product is even

\(\Rightarrow\) necessarily even.

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