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Question

Difficulty: EasyCounting with Restrictions and Repetitions

How many distinct 4-letter arrangements can be formed by rearranging all of the letters in the word SEES\text{SEES}?

Answer: 6

Answer

6
To calculate the number of unique arrangements for a multiset of letters, compute the total factorial divided by the product of factorials for each repeated letter's frequency. For SEES\text{SEES}, there are 4 total letters with 2 'S's and 2 'E's, resulting in 4!2!2!=244=6\frac{4!}{2!2!} = \frac{24}{4} = 6.

Step-by-Step Solution

1
Count the total number of letters and identify frequencies of repeated letters.
The word SEES\text{SEES} contains 4 letters in total: two 'S's and two 'E's.
To apply the distinct permutations formula, we need the total count of elements and the counts for each repeated identical element.
2
Calculate the number of distinct arrangements using the formula n!n1!n2!…nk!\frac{n!}{n_1! n_2! \dots n_k!}.
\frac{4!}{2! \cdot 2!} = \frac{24}{2 \cdot 2} = 6.
Dividing by 2!⋅2!2! \cdot 2! eliminates duplicate counts arising from swapping indistinguishable identical letters.

Key Concept

Permutations with Repetition
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Topics

Counting with Restrictions and Repetitions

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