[Math] How many combinations of four letters each can be made from the word PEPPER

combinationscombinatoricsprobability

As the title says. I know that if there is no certain number of letters to choose you would have to just do 6!/3!2!. But what would you do if you have to only choose 4 letters?

Best Answer

My answer comes rather late, so I don't expect up votes.

The response by user155957 was based on assuming that when you said 'combinations', you really meant 'permutations'. Based on this assumption, that user's response is correct by my reckoning.

Let us assume that when you said 'combinations', you really meant combinations.

In that case, the answer is 5.

If all the letters were distinct, such as they are in 'PENCIL', then the number of combinations possible with 6 distinct letters, taking 4 at a time, is given by the function nCr( 6, 4), which gives 15. However, the word is 'PEPPER', and the letters are not all distinct.

Writing the 'source partition' for the word PEPPER gives:

 P E R
[3,2,1]  <-- source partition

For combinations of 4 letters, we write all three possible 'target partitions' that go with [3,2,1]. These are:

[3,1]
[2,2]
[2,1,1]

Next to these, we list the corresponding selections (which are combinations, NOT permutations!)

[3,1]    PPPE, PPPR
[2,2]    PPEE
[2,1,1]  PPER, EEPR

In conclusion, there are 5 combinations of 4 letters each that can be made from the word PEPPER.

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