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The full question is: How many ways can the 60 people from 20 countries be seated around a table so that each president, vice president, and poet laureate are sitting consecutively?

The answer I come up with is : (19!)*6

19! = the number of ways to arrange each country in the table = (20-1)!

6 = the number of ways to arrange the seat of three people in each country which are:

[P,V,L]

[P,L,V]

[V,P,L]

[V,L,P]

[L,V,P]

[L,P,V]

But I'm an not sure if whether or not this answer is correct. Please help me. Thank you

JTJung
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2 Answers2

2

You are nearly correct. Let's look at the steps needed to create a valid arrangement of people:

Step 1: Order the countries. $19!$ orderings (correct).

Step 2: Order the three people from country one. $6$ orderings (correct).

Step 3: Order the three people from country two. $6$ orderings (important).

...

Step 21: Order the three people from country $20$. $6$ orderings (important).

By the product rule we then have a total of $19!6^{20}$ arrangements.

DreamConspiracy
  • 1,195
0

Consider, each triplet of [P, V, L] as one bundle. So, there are total $20$ bundles.

Now, arranging these $20$ bundles on a circular table requires $(20-1)!$

Also, these bundles can be arranged internally which requires $3!$

Thus, the answer will be $$19! * (3!)^{20}$$

AMAN
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