Multiple choice

The number of ways in which the letters of the word $ARRANGE$ can be permuted such that the $R$'s occur together is

  1. $\dfrac{7!}{2!2!}$
  2. $\dfrac{7!}{2!}$
  3. $\dfrac{6!}{2!}$
  4. $5! \times 2!$
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C Correct answer
Explanation

Word ARRANGE has 7 letters: A:2, R:2, N:1, G:1, E:1. Treating RR as one unit, we have {RR, A, A, N, G, E}, which is 6 items. Permutations = 6! / 2! (for the two A's).

AI explanation

The word ARRANGE contains 7 letters. To keep the two R's together, we treat them as a single combined block. This gives us 6 items to arrange (the RR block, two A's, and three other distinct letters). Using the permutation formula for a multiset, the total arrangements are 6! divided by 2! to account for the repeated A's. This yields the result 6! / 2!.

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