15 persons among whom are Sally and Don sit down at random at a round table.The probability that there are 4 persons between Sally and Don is?

This is how i solved it:

Total possible outcomes: $14!$

Favourable outcomes: Lets fix Sally's position, then 4 positions next to her can be filled in $4!$ ways and then after Don's fixed position, the remaining $9$ people can be seated in $9!$ ways. So, Favourable outcomes : $2\times1\times4!\times1\times9!$

So, Probability: $$\frac{ 2\times1\times4!\times1\times9! }{14!}$$

Is this correct?


This is incorrect, because the number of ways to choose the four people between Sally and Don is $13\times12\times11\times10$ (they can be any four of the other thirteen, so it's not just ordering four specific people).

The easy way to do this is that once Sally is fixed there are $2$ out of $14$ positions where Don could sit for this to be true, so the probability is $\frac2{14}=\frac17$.


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