Permutations With Repetition

Question

The number of distinct arrangements that can be formed using all the letters of the word BANANA, accounting for repeated letters, equals which value?

RankGuru · Accepted Answer

When some objects are identical, the number of distinct permutations is the total factorial divided by the factorials of the repetition counts, a core JEE Advanced formula. The word BANANA has 6 letters: A appears 3 times, N appears 2 times, and B appears once. The number of distinct arrangements is 6! divided by (3! × 2! × 1!) = 720 / (6 × 2 × 1) = 720 / 12 = 60. Dividing by the factorials of repeated letters removes the overcounting caused by permuting identical letters among themselves. Option 720 = 6! treats all letters as distinct. Option 120 = 5! mishandles the repetition. Option 360 divides by only one repetition factor. Hence there are 60 distinct arrangements. Plausibility check: the count must be smaller than 6! = 720 by exactly the factor 3! × 2! = 12 that accounts for indistinguishable rearrangements, and 720/12 = 60 confirms the multinomial correction.

Permutations With Repetition

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