Question

11. In how many different ways can the letters of the word MATH be rearranged to form a four-letter code word (i.e. it doesn't have to be a word in English)?

Answer 1

The number of letters in the word MATH IS

`=4`

To rearrange a letter with n letters to form a four-letter code will be

`\begin{gathered} n! \\ \text{Where n=4} \end{gathered}`

Hence,

The number of ways of rearranging the words will be

`\begin{gathered} =4! \\ =4*3*2*1 \\ =24\text{ ways} \end{gathered}`

This problem is a bit different. Instead of choosing one item from each of several different

categories, we are repeatedly choosing items from the same category (the category is: the

letters of the word MATH) and each time we choose an item we do not replace it, so there is

one fewer choice at the next stage: we have 4 choices for the first letter (say we choose A),

then 3 choices for the second (M, T, and H; say we choose H), then 2 choices for the next

letter (M and T; say we choose M) and only one choice at the last stage (T). Thus, there are

4 · 3 · 2 · 1 = 24 ways to spell a code word with the letters MATH.

Hence,

The final answer = 24 ways