Permutations of Words with Repeated Letters

Introduction to Permutations

In combinatorics, a permutation is an arrangement of items in a specific order. When we have a set of $n$ distinct items, there are $n!$ ways to arrange them. However, when some items are identical (like the repeating letters in a word), the number of distinct arrangements is significantly reduced.

The Formula

The formula for the number of distinct permutations of $n$ objects, where $n_1$ are of one type, $n_2$ are of another type, and so on, is:

$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Solving for 'MATHEMATICS'

To find the number of permutations of the word 'MATHEMATICS', we follow these steps:

1. Count the total letters ($n$): The word M-A-T-H-E-M-A-T-I-C-S has 11 letters.

2. Identify repeated letters:

   • M appears 2 times.
   • A appears 2 times.
   • T appears 2 times.
   • H, E, I, C, S each appear 1 time.

3. Apply the formula: $$\text{Total Permutations} = \frac{11!}{2! \cdot 2! \cdot 2!}$$

4. Calculate:

   • $11! = 39,916,800$
   • $2! \cdot 2! \cdot 2! = 2 \cdot 2 \cdot 2 = 8$
   • Result = $39,916,800 / 8 = 4,989,600$

There are 4,989,600 distinct ways to arrange the letters in 'MATHEMATICS'.