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

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Prepare for the AMSOC 26-003 Module A Test. Utilize flashcards and multiple choice questions with hints and explanations. Ace your exam!

Multiple Choice

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

Explanation:
When counting distinct arrangements of letters with repeats, you start with the total number of positions and then divide out the duplicates caused by identical letters. BALLOON has seven letters with two Ls and two Os, so if all seven were different there would be 7! = 5040 permutations. But swapping the two Ls doesn’t create a new arrangement, and the same with swapping the two Os. That means we divide by 2! for the Ls and by 2! for the Os. So the total is 5040 / (2 × 2) = 1260. This matches the correct count of distinct permutations.

When counting distinct arrangements of letters with repeats, you start with the total number of positions and then divide out the duplicates caused by identical letters. BALLOON has seven letters with two Ls and two Os, so if all seven were different there would be 7! = 5040 permutations. But swapping the two Ls doesn’t create a new arrangement, and the same with swapping the two Os. That means we divide by 2! for the Ls and by 2! for the Os. So the total is 5040 / (2 × 2) = 1260. This matches the correct count of distinct permutations.

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