Question

In how many ways can 9 examination papers be arranged so that the best and the worst papers never come together?

Answer 1

We will solve this question by using the formula: the ways when the best and the worst papers are not together = the total number of ways to arrange 9 papers – the ways in which the best and the worst papers are together. The total number of ways will be given by n! , where n is the number of total examination papers.

Complete step-by-step answer:
In the question, we are asked to calculate the total number of ways in which 9 papers can be arranged such that the best and the worst papers are never together.
For this, we know that the ways can be calculate by using the formula:
The ways when the best and the worst papers are not together = the total number of ways to arrange 9 papers – the ways in which the best and the worst papers are together
We will calculate the total number of ways to arrange the papers by the formula: n!
$\Rightarrow n! = 9!$
And we know that n! is written as: (n) $\times$ (n – 1) $\times$ (n – 2) $\times$ (n – 3) $\times$…$\times$3$\times$2$\times$1
$\Rightarrow 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362,880$
Now, for calculating the ways when the best and the worst papers are together, we will assume that the best and the worst paper to be one paper and hence the total number of papers will be 8 now.
These 8 papers can be arranged in 8! Ways and now we can tell that the best and the worst paper which we considered as one can be arranged in 2! Ways on their own.
Therefore, the total number of ways will be 8! $\times$2!
Now, $8! \times 2! = \left( {8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} \right) \times \left( {2 \times 1} \right)$
$\Rightarrow 8! \times 2! = \left( {40,320} \right) \times \left( 2 \right) = 80,640$
On putting these values in the formula:
The ways when the best and the worst papers are not together = the total number of ways to arrange 9 papers – the ways in which the best and the worst papers are together
$\Rightarrow$ The ways when the best and the worst papers are not together = 362,880 – 80,640 = 282,240.

Note: In this question, you may get confused in the main formula because you need to think about the total possible ways when the two papers (best and the worst) aren’t together and there is no direct formula. Be careful while calculating the total number of ways when the two papers are together by assuming them as one because you may forget that both of them can also be arranged by switching their places and resulting in a different value.