Permutations in a conference

[Jshua Monkoe]

Homework Statement

At a conference of 5 powers,each deligation consists of 3 members. If each delegation sits together, with the leader in the middle, in how many ways ca the members be arranged at a round table?

Homework Equations

No. of ways of arranging n objects around a circle=(n-1)!
P(n,r)=n!/(n-r)!

The Attempt at a Solution

I understand that 15 people go around the table
=> no. of ways to arrange=14!
But again in their different delgations,each can be arranged 2!/(2-2)!=2ways
=> There are 2(15) ways to arrange within the delegations
.'. there are 14!(30) ways
IS THIS CORRECT?

 

[Abha]

We can not consider 15 people as a whole since we do not have complete flexibility regarding their arrangement.
Since there are 5 groups which always individually sit together, there are 4! ways in which the groups can be organized on the table.
Now, each group can internally have 2 possible arrangements. => 2^5 arrangements in all.
i.e. Total no. of permutations = 4! * 2^5

 

[Architeuthis Dux]

Your attempt at a solution is close, but not entirely correct. First, we need to clarify that there are actually 6 delegations (not 5) since each delegation consists of 3 members. So, there are 6 sets of 3 members each, making a total of 18 people at the conference.

Now, we can use the formula for arranging n objects around a circle, which is (n-1)!. In this case, n=18, so the number of ways to arrange the 18 people around the table is (18-1)! = 17!.

However, we also need to account for the fact that each delegation sits together with the leader in the middle. This means that each delegation can be arranged in 3! ways (since there are 3 members in each delegation). So, we need to multiply 17! by (3!)^6 (since there are 6 delegations) to get the total number of arrangements.

Therefore, the final answer is 17!(3!)^6 = 17! * 729 = 5,467,885,056,000 ways.

In general, when arranging objects around a circle and also arranging objects within a set, we need to use the formula P(n,r) = n!/(n-r)! to account for both arrangements. In this case, n=18 and r=3, so we would have P(18,3) = 18!/(18-3)! = 18!/15! = 18*17*16 = 5,472 ways for each delegation to be arranged. Then, we multiply this by the number of delegations (6) to get the total number of arrangements, which is 5,472 * 6 = 32,832. This is a significantly smaller number than the previous answer, so it is important to make sure we are using the correct formulas and understanding the problem correctly.