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How many distinct 4-letter words (with or without meaning) can be ma [#permalink]
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olaflesniak1998 wrote:
Could you maybe elaborate why you in case 2 and 3 multiply by factorial 4! ?


We have the following combinations of alphabets available with us -

E X M T O A A I I N N

Question: Create four-letter words with or without meaning.

The creation of a letter involves the following two steps -

Step 1: Select four letters

Step 2: Arrange those four letters

Case 1: No letters repeat (4 distinct alphabets)

Step 1: Select four letters

There are 8 distinct letters, hence we can select 4 letters in \(^8C_4\) ways.

Step 2: Arrange those four letters

The 4 distinct letters so selected in step 1, can be arranged in 4! ways.

Possible words : \(^8C_4\) * 4! = 1680

Case 2: One pair of alphabet repeats (2 distinct alphabets & 1 repeating pair)

Step 1: Select four letters

The repeating pair can be selected in \(^3C_1\) ways

From the remaining 7 distinct alphabets, we can select two alphabets in \(^7C_2\) ways

Step 2: Arrange those four letters

The word is of the form XXYZ, and the number of possible arrangements = \(\frac{4!}{2!}\)

XX is the repeating pair
Y & Z are unique alphabets

Possible words : \(^3C_1 * ^7C_2 * \frac{4!}{2!}\) = 756

Case 3: Two pairs of alphabet repeats (2 pairs of repeating alphabets)

Step 1: Select four letters

The repeating pairs can be selected in \(^3C_2\) ways

Step 2: Arrange those four letters

The word is of the form XXYY, and the number of possible arrangements = \(\frac{4!}{2!*2!}\)

XX & YY are the repeating pairs

Possible words : \(^3C_2 * \frac{4!}{2!*2!}\) = 18

Total of all three cases = 1680 + 756 + 18 = 2454

Option B

Hope this clarifies your question!
GMAT Club Bot
How many distinct 4-letter words (with or without meaning) can be ma [#permalink]
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