Question

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

210
1,260
2,520
5,040

Answer 1

To find the number of ways to arrange the letters in the word "balloon", we first need to identify the total number of letters and any repetitions.

The word "balloon" consists of 7 letters in total:
- b: 1
- a: 1
- l: 2
- o: 2
- n: 1

We can use the formula for permutations of a multiset to find the number of arrangements. The formula is given by:

\[
\frac{n!}{n_1! \times n_2! \times n_3! \times \ldots}
\]

where \(n\) is the total number of letters, and \(n_1, n_2,\) etc. are the frequencies of the repeated letters.

For "balloon":
- Total letters \(n = 7\)
- Repeats: \(l\) appears 2 times and \(o\) appears 2 times.

The calculation is as follows:

\[
\text{Number of arrangements} = \frac{7!}{2! \times 2!}
\]

Calculating \(7!\) and \(2!\):
\[
7! = 5040
\]
\[
2! = 2
\]

Now substitute these values into the formula:

\[
\frac{7!}{2! \times 2!} = \frac{5040}{2 \times 2} = \frac{5040}{4} = 1260
\]

Thus, the number of ways to arrange the letters in the word "balloon" is \( \boxed{1,260} \).