A group of 9 friends go into a nightclub and there is only a table for 3 and therefore the others will have to stand up. If the friends randomly assign who sits at the table, in how many ways can they be seated?

Question

A group of 9 friends go into a nightclub and there is only a table for 3 and therefore the others will have to stand up. If the friends randomly assign who sits at the table, in how many ways can they be seated?

These permutation and combination q's are giving me such a hard time Thanks so much to anyone who helps. Answer is 252

Answer 1

First let's choose any 3 of the 9 friends to place at the table

which would be C(9,3) or 84
Now it depends on what we consider the seating arrangement at the
table.
If the seats are specified, e.g. one facing against a wall , etc
the each of the groups of 84 triples can be arranged in 3! or 6 ways.
So the number of seatings is 84*6 or 504
However, suppose everybody stands up and moves one seat to the left.
Is the seating arrangement still the same? e.g. for a round table it would still
be the same.
The given answer seems to suggest that. So let's place one of the chosen
people at the table, then the other 2 can be seated in 2*1 or 2 ways.

So.... number of ways = 84*3! /2 = 252

I agree, it is a rather tricky one.

Answer 2

To solve this problem, we can use the concept of permutations and combinations.

First, let's consider the number of ways the friends can be seated at the table. Since there are only 3 seats available, we need to choose 3 friends out of the 9 to sit. This can be done in a specific order, so we use permutations.

The number of ways to select 3 friends out of 9 would be represented as "9P3" or "Permutation of 9, taken 3 at a time." This can be calculated using the formula:

nPr = n! / (n - r)!

Here, n represents the total number of friends (9) and r represents the number of friends seated at the table (3).

So, substituting the values into the formula:

9P3 = 9! / (9 - 3)!
= 9! / 6!
= (9 * 8 * 7 * 6!) / 6!
= 9 * 8 * 7
= 504

However, this only represents the number of ways the friends can be seated at the table. To find the total number of ways they can be seated (which includes both seated and standing positions), we need to consider the remaining 6 friends who will be standing.

The number of ways the remaining 6 friends can stand can be calculated as "6!" or "Factorial 6."

So, multiplying the number of seating arrangements by the number of standing arrangements, we get:

504 * 6! = 504 * 6 * 5 * 4 * 3 * 2 * 1 = 30240

Therefore, there are a total of 30,240 ways the friends can be seated and standing.