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ISI 2019 | PCB Mathematics | Question: 21

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Consider all the permutations of the numbers $1,2, \ldots, 9$. Find the number of permutations which satisfy all of the following:

  • the sum of the numbers lying between $1$ and $2\;($including $1$ and $2)$ is $12,$
  • the sum of the numbers lying between $2$ and $3\; ($including $2$ and $3)$ is $23,$
  • the sum of the numbers lying between $3$ and $4\;($including $3$ and $4)$ is $34,$
  • the sum of the numbers lying between $4$ and $5\; ($including $4$ and $5)$ is $45.$
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First we'll try to find positions or relative positions of $1, 2, 3, 4, 5$ and for remaining numbers, we'll see what can be done with them later.

Sum (inclusive) of numbers lying between $4 \text{ and }5 = 45$.

$\sum_{i=1}^{9} i = \frac{9(9+1)}{2} = 45$.

$\therefore 4,5$ must be in extreme corners.

Sum (exclusive) of numbers lying between $4 \text{ and }5 = 45 - 4 - 5 = 36$.

$[4, sum(36), 5]$.

Sum (inclusive) of numbers lying between $3 \text{ and }4 = 34$.

Sum (exclusive) of numbers lying between $3 \text{ and }4 = 34 - 3 - 4 = 27$.

Sum (exclusive) of numbers lying between $3 \text{ and }5 = 45 - 34 - 5 = 6$.

$[4, sum(27), 3, sum(6), 5]$.

Sum (inclusive) of numbers lying between $2 \text{ and }3 = 23$.

Sum (exclusive) of numbers lying between $2 \text{ and }3 = 23 - 2 - 3 = 18$.

Sum (exclusive) of numbers lying between $2 \text{ and }4 = 34 - 23 - 4 = 7$.

$[4, sum(7), 2, sum(18), 3, sum(6), 5]$.

Sum (inclusive) of numbers lying between $1 \text{ and } 2 = 12$.

Sum (exclusive) of numbers lying between $1 \text{ and } 2 = 12 - 1 - 2 = 9$.

Sum (exclusive) of numbers lying between $1 \text{ and } 3 = 23 - 12 - 3 = 8$.

$[4, sum(7), 2, sum(9), 1, sum(8), 3, sum(6), 5]$.

$\therefore$ possible permutations are - 

$4, 7, 2, 9, 1, 8, 3, 6, 5 \text{ and } 5, 6, 3, 8, 1, 9, 2, 7, 4$.

Answer :- 2.

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