Question

In how many ways can the letters in the word spoon be arranged?

24
30
60
120

Answer 1

To find the number of ways to arrange the letters in the word "spoon," we first note that the word contains 5 letters, with the letter 'o' appearing twice.

The formula for the number of unique arrangements of letters in a word is given by:

\[
\frac{n!}{n_1! \times n_2! \times \ldots \times n_k!}
\]

where:
- \( n \) is the total number of letters,
- \( n_1, n_2, \ldots, n_k \) are the frequencies of the repeated letters.

In our case:
- The total number of letters, \( n = 5 \) (s, p, o, o, n).
- The letter 'o' appears twice, so \( n_1 = 2 \) for 'o'.

Thus, we can apply the formula as follows:

\[
\text{Number of arrangements} = \frac{5!}{2!}
\]

Now calculating \( 5! \) and \( 2! \):

\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]

\[
2! = 2 \times 1 = 2
\]

Now substitute these values back into the formula:

\[
\text{Number of arrangements} = \frac{120}{2} = 60
\]

Therefore, the total number of ways to arrange the letters in the word "spoon" is \(\boxed{60}\).