How many distinct permutations are there of the letters in 'BALLOON'?

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Multiple Choice

How many distinct permutations are there of the letters in 'BALLOON'?

Explanation:
When some letters repeat, you have to account for the indistinguishable swaps of those repeats. BALLOON has seven letters in total, with two Ls and two Os that are identical. If every letter were different, there would be seven factorial arrangements. But because the two Ls are the same and the two Os are the same, each arrangement has been counted twice for the Ls and twice for the Os. So you divide by two factorial for the Ls and by two factorial for the Os. That gives seven factorial divided by (two factorial times two factorial) = 5040 / 4 = 1260 distinct permutations.

When some letters repeat, you have to account for the indistinguishable swaps of those repeats. BALLOON has seven letters in total, with two Ls and two Os that are identical. If every letter were different, there would be seven factorial arrangements. But because the two Ls are the same and the two Os are the same, each arrangement has been counted twice for the Ls and twice for the Os. So you divide by two factorial for the Ls and by two factorial for the Os. That gives seven factorial divided by (two factorial times two factorial) = 5040 / 4 = 1260 distinct permutations.

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