# Combining Letters into Words

It is possible to use the Law of Combining even if the two "sets" are actually identical. For example, you might want to know how many two letter words there are in the English language (or you might not, but humor me here).

To simplify matters, let's assume that ANY combination of two letters is a word, regardless of whether it contains a vowel or not.

In this case, "set 1" contains all the letters in the English alphabet. So does "set 2".

So the number of words is:

Think for a minute about these questions:

1. Does it matter what order you pick the letters?

2. Does it count if you pick the same letter twice?

A popular children's game involves making new words from a long word or phrase. For example, you could start with BIOLOGY FOR FUN AND PROFIT and make NAP, FAN, BART and BRAT. One way to generate new words is simply to try all possible combinations of letters.

### How many 5-letter words could be made?

(To make this problem interactive, turn on javascript!)

• I need a hint ... : Although the phrase contains 22 letters, several are repeated. There are only 14 distinct letters.

#### I think I have the answer: 537,824 =(14*14*14*14*14)

Sometimes hitting the multiplication key so many times gets to be a bit tedious...

### How many 9 letter combinations are possible?

(To make this problem interactive, turn on javascript!)

• I need a hint ... : typing 14*14*14*14*14*14*14*14*14 into Google (or a calculator) is boring... there must be an easier way.

• ...another hint ... : multiplying something by itself 9 times is the same as raising to the 9th power.

• ...another hint ... : for Google: Use the "^" key to get a power, like this: "14^9="

#### I think I have the answer: 2.0 × 1010

This is a good trick to remember: if you are making a string of choices from the same set, then instead of multiplying repeatedly, you can simply use the a power. If we want to make a 9 letter word, and we have 14 letters available to us, we can do this:

14*14*14*14*14*14*14*14*14 = 2.0 × 1010

to figure out how many possible words we could make, or we could simply do this:

14^9 = 149 = 2.0 × 1010

The first way is a little easier to understand -- it almost looks like a 9-"letter" word. But the second way is a lot faster to calculate.