Question

Count the number of ways to seat 34 people, consisting of 17 couples, in a row of 34 seats if each couple must get adjacent seats.

Answer 1

2^17 × 17!

Explanation:

Seat 1 = 34 possibilities

Seat 2 = 1 possibility

Seat 3 = 32 possibilities

Seat 4 = 1 possibility

And so on

So it is 34 × 32 × 30 × ... × 2

So it is

`{2}^(17) * 17factorial`

I hope this helps you :)

Answer 2

4.6620663x10^19 or 46,620,663,000,000,000,000 total configurations

Explanation:

Since each couple must get adjacent seats, we can first think of the row as having 17 seats instead of 34 since the people in each pair does not change and we consider instead the position of the pair rather than the two people.

There are 17 places where each couple can sit in the 34 chairs. Therefore the number of ways to order the 17 pairs (regardless of the order of individual people in those pairs) is 17!.

This is because first place could be taken by any of the 17 couples so there are 17 possibilities. After the first place is taken, there are only 16 possible couples to take the second place to we have 17x16. After the first two are taken there are only 15 possible couples to take the third place so we multiply by 15. This continues with each respective place so the total number of configurations is: 17x16x15x14x13x12x11x10x9x8x7x6x5x4x3x2x1 OR 17! which is equal to 3.5568743x10^14, or 355,687,430,000,000 different configurations.

Now we must also consider the order of the individual people in the pairs as one can sit on the left or on the right of the other person (e.g. 1,2 or 2,1). And since each pair can have two possible orders, every possible configuration of the pairs in the 17 places must be multiplied by 2^17 to accommodate for this (2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2 for the 17 couples), which is equal to: 131072

Therefore the total number of ways to seat 17 couples in a row of 34 is 131072x355,687,430,000,000= 4.6620663x10^19 or 46,620,663,000,000,000,000 total configurations.

Hope this helped!