# Emma's Dilemma

Emma’s Dilemma

The Problem

Emma is looking at the different arrangements of her name.

1)        Find the different arrangements of Emma’s name.

2)        Emma has a friend named Lucy; find the different arrangements of her         name.

3)        Investigate the number of different arrangements of the letters of names         you have chosen.

Investigating the Problem

To investigate this problem I am going to look at the different arrangements of a four letter name with two letters the same and a four letter name with all different letters. I am then going to try three and five letter names, with two letters the same and all different. I am then going to look at the results and predict the next set. I will then move on to three and four letters the same.

Four Letter Words

One/Two Letter Words

We know that a one letter word will only have 1 combination. For two letters we know that it only has 2 different combinations and 1 for both letters repeating:

1. A

1. AB
2. BA

1. AA

Three Letter Words

Five Letter Words

James is a five letter word with different letters. Without writing down all the different combinations we can find them out:

J(AMES)        By putting the J at the front of the word, we are effectively changing a four letter word with all different letters – AMES. There are 24 different combinations of a four letter word with no repeating letters. By multiplying this by 5 we have the result for a five letter word with no repeats.

5                                X                                                24

Number of letters in James                        Number of combinations for a four letter word

This totals to 120, so a five letter word with no repeats has 120 different combinations. By applying this rule and using previous results, we can find out the number of combinations for a particular number of letters without writing them all out.

Results

From the results we can already see a pattern. The combination for all different letters divided by two equals that of two letters the same in a word.

Prediction

By looking at the results I have ...