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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



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


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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