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**fusilli_jerry89****Member**- Registered: 2006-06-23
- Posts: 86

Determine the total number of possible "words" using at least one of the letters of B-A-R-H-A-M.

I know how to get all the 6 letter words, 6!/2!, and the 1 letter words, obviously 5, but is there an easier way to get all the 2,3,4 and 5 letter words instead of just counting?

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,632

Maybe Combinations and Permutations and Combinations and Permutations Calculator will help you.

But be careful of duplicate words, because there are two "a"s!

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**pi man****Member**- Registered: 2006-07-06
- Posts: 251

You're right on the 6 letter words.

Let's do 5 letter words first. You're concerned about 2 groups of letters; those groups without duplicates and those with duplicates (the "a"). There is only one group of 5 letters without duplicates. You can arrange those 5! different ways or 120.

Now let's consider the 5 letter groups with 2 a's. You need 3 more letters from the remaining four, i.e. (4 choose 3) which is 4. You can arrange those groups of 5 letters in 5! /2! different ways (60).

So the number of 5 letter words is 120 + (4 * 60) = 360.

4 letter words:

Number of words without duplicates: Choose 4 letters from the possible 5 unique letters (5 different ways) and multiple by the number of ways you can arrange them (4!). That's 5*24 = 120.

Number of words with duplicate a's: Use 2 a's plus 2 of the remaining 4 (4 choose 2). You can arrange those groups 4!/2! ways. That's 6*12 = 72. Add that to the 120 and you have 192 four letter words.

3 letter words:

Number of words without duplicates: Choose 3 letters from the possible 5 unique letters (10) and multiple by the number of ways you can arrange them (3!). That's 10*6=60

Number of words with duplicate a's: Use 2 a's plus 1 of the remaining 4 (4 choose 1). You can arrange those groups 3!/2! ways. That's 4*3 = 12. Add that to the 60 and you have 72 three letter words.

2 letter words:

Number of words without duplicates: Choose 2 letters from the possible 5 unique letters (10) and multiple by the number of ways you can arrange them (2!). That's 10*2=20

Number of words with duplicate a's: Just 1 (aa). Add that to the 20 and you have 21 two letter words.

Add them all together and you have your answer.

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