How many distinct 5-letter code words can be formed by rearranging all of the letters in t… — GMAT

How many distinct 5-letter code words can be formed by rearranging all of the letters in the word $\text{LEVEL}$ ?

A
$15$
B
$24$
$30$ Answer
D
$60$
E
$120$

Answer

The total number of distinct 5-letter code words that can be formed is 30.

To calculate the number of unique permutations of a set containing repeated items, divide the factorial of the total number of items by the product of the factorials of the counts of each repeated item. In the word

\text{LEVEL}

, there are 5 letters total, with the letter 'L' appearing 2 times and the letter 'E' appearing 2 times. The calculation is

\frac{5!}{2! \times 2!} = \frac{120}{4} = 30

Step-by-Step Solution

Identify the total number of letters and the counts of identical letters.

The word

\text{LEVEL}

has 5 total letters, where 'L' appears 2 times and 'E' appears 2 times.

Duplicate elements produce identical arrangements if treated as distinct items.

Apply the permutation formula for repeated items:

\frac{n!}{n_1! \times n_2! \times \dots}

\frac{5!}{2! \times 2!} = \frac{120}{2 \times 2} = \frac{120}{4} = 30

Dividing

5!

2!

for the repeated 'L's and

2!

for the repeated 'E's eliminates duplicate arrangements.

Key Concept

Permutations with Repetition / Indistinguishable Objects

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