Question

how many ways can we arrange 4 boys around a circular table​

Answer 1

Step-by-step explanation:

There are 7 ways to choose a boy for Chair 1, then 6 ways for Chair 2 down to 1 for Chair 7, which leaves us with

7×6×5×4×3×2×1=7!=5040

combinations.

But your specific question is not about numbered chairs, but about chairs arranged in a circular manner: there are two ways to understand that mathematically.

First scenario, what you mean is the arrangement of boys clockwise around the table. In this case, the chairs are ordered but not numbered: we can start counting them starting from either one of them, then arrange them in ascending starting from there. In other way, the arrangement is the same whether we start counting at Chair 1, Chair 3 or Chair 6. So, in this scenario, the number of combinations is the previous number divided by 7 , as we have 7 equivalent dispositions for the same arrangement:

50407=720

Second scenario, we also do not mind if the boys are arranged clockwise or counter-clockwise. Which means, what you mean by “ways can 7 boys be seated” really means “who are the two neighbors of each boy”. In that case, we ha

pve two equivalent arrangements in our previous scenario for each valid arrangement of this scenario (clockwise and counter-clockwise), which means we have to divide the number of previous arrangements by 2 :

7202=360

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