Question

in how many ways can the letters in the word balloon be arranged?
210
1,260
2,520
5,040

Explanation:

Step1: Count letters and duplicates

The word "balloon" has 7 letters. Letters: b, a, l, l, o, o, n. Duplicates: l (2 times), o (2 times).

Step2: Use permutation formula for duplicates

The formula for permutations of a word with repeated elements is $\frac{n!}{n_1!n_2!...n_k!}$, where $n$ is total letters, $n_i$ are counts of each repeated letter. Here, $n = 7$, $n_1 = 2$ (for l), $n_2 = 2$ (for o). So calculate $7! = 5040$, $2! = 2$, so $\frac{7!}{2!×2!} = \frac{5040}{2×2} = 1260$.

Answer:

1,260