## Monday, April 5, 2010

### Permutations and Combinations - The Answers!

Dear Friends:

Here are the answers (long-awaited) for the questions posed las week which involve permutations and combinations:

1. In a group of ten individuals, how many pairings are possible?

Based upon the formula (you might want to roll back a few posts to find it), the answer is 10!/(10! - 2!)(2!), which equals 10 x 9 /2 which equals 45 possible pairings.

2. In a group of ten individuals, how many combinations of 3 individuals each are possible?

Using the same formula as above, the answer is 10 x 9 x 8 / 3, which equals 168 possible combinations of 3 each.

3. In a group of ten individuals, how many combinations of 2 or 3 individuals are possible?

The key word is "or." Merely add the answer to question 1 to the answer to question 2, or simply add 45 + 168, which equals 213 possible combinations of either 2 or 3 each.

4. Five paintings are to be arranged in a row on a wall. How many different arrangements are possible?

Using the permutation formula, the answer can be found by multiplying 5 x 4 x 3 x 2 x 1, which will equal 120 possible arrangements.

5. Same facts as in question 4, above: How many different arrangements are possible where two paintings by one particular artist must be side by side?

If one artist produced two fo the paintings, and he or she would like both of those paintings displayed side by side, we will have less than 120 possible arrangements. There are several arrangements possible where two paintings MUST be side by side:

1 x 1 x 3 x 2 x 1 =  6 arrangements

3 x 1 x 1 x 2 x 1 =  6 arrangements

3 x 2 x 1 x 1 x 1 =  6 arrangements

3 x 2 x 1 x 1 x 1 =  6 arrangements

In all, there are 6 + 6 + 6 + 6 total arrangements, or a total of 34 possible arrangements where the demanding artist will have his or her two works hanging side by side.

Faithfully,

Douglas Castle