Arrangements

Probability Level 2

How many ways are there to arrange the letters of the word R E C U R R E N C E RECURRENCE in a row so that no two R R `s are adjacent?


The answer is 23520.

Michael Huang by Michael Huang , 40, China

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1 solution

Mark Hennings
May 28, 2019

There are:

  • 10 ! 3 ! × 3 ! × 2 ! = 50400 \frac{10!}{3! \times 3! \times 2!} = 50400 permutations of RECURRENCE in total,
  • 8 ! 3 ! × 2 ! = 3360 \frac{8!}{3! \times 2!} = 3360 permutations of RECURRENCE where all three Rs are adjacent,
  • 2 ( 9 ! 2 ! × 3 ! × 2 ! − 8 ! 3 ! × 2 ! ) = 23520 2\left(\frac{9!}{2! \times 3! \times 2!} - \frac{8!}{3! \times 2!}\right) = 23520 permutations of RECURRENCE where two Rs are adjacent, but the third R is separate

Thus there are 50400 − 3360 − 23520 = 23520 50400 - 3360 - 23520 = \boxed{23520} permutations of RECURRENCE where no two Rs are adjacent.

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