in how many ways can the letters in the word balloon be arranged?\no 210\no 1,260\no 2,520\no 5,040

Explanation:

Step1: Count total letters and repeated ones

The word "balloon" has 7 letters. The letter 'l' appears 2 times and the letter 'o' appears 2 times.

Step2: Use permutation formula for repeated - elements

The formula for permutations of a word with (n) total elements and (n_1,n_2,\cdots,n_k) repeated elements is (\frac{n!}{n_1!n_2!\cdots n_k!}). Here (n = 7), (n_1=2) (for 'l') and (n_2 = 2) (for 'o'). So the number of arrangements is (\frac{7!}{2!2!}).

Step3: Calculate factorial values

We know that (n!=n\times(n - 1)\times\cdots\times1). So (7!=7\times6\times5\times4\times3\times2\times1 = 5040), (2!=2\times1=2). Then (\frac{7!}{2!2!}=\frac{5040}{2\times2}).

Step4: Perform the division

(\frac{5040}{4}=1260).

Answer:

1,260