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How many ways we can arrange 4 letters from PROFESSOR?

The way I tried to do is by grouping the repeated words like (OO), (RR), (SS) and that can be done in three ways which seems to me a bit complicated and time consuming too. Can't we do this any simple way that wouldn't take much time?

SchrodingersCat
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Rayan Ahmed
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    http://math.stackexchange.com/questions/1562270/in-how-many-ways-can-we-arrange-4-letters-of-the-word-engine?rq=1 – Mr. Y Dec 12 '15 at 17:10

2 Answers2

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E, F, OO, P, RR, SS

Break it into cases:

  • No letter is repeated: Pick which four letters they are and in what order they appear: $6\cdot 5\cdot 4\cdot 3$ possibilities.

  • Exactly one letter is repeated: Pick which letter it is that is repeated ($3$ choices), pick which two remaining letters are used ($\frac{5\cdot 4}{2}$ choices), and pick how they are arranged ($\frac{4!}{2!}$)

  • Exactly two letters are repeated: Pick which two letters are repeated ($3$ choices), pick the order they appear $(\frac{4!}{2!2!})$

This gives a total of $\frac{6!}{2!}+3\cdot 5!+3\cdot 3!=738$ by my count.

JMoravitz
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HINT: You can have $3$ cases:

  1. All $4$ different letters
  2. $2$ letters same and $2$ different letters
  3. $2$ pairs of same letters, each pair being distinct from the other

Consider this and calculate accordingly.

SchrodingersCat
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