In how many different ways can the letters of the word "LEVEL" be arranged?

Start with 5! = 120, then just keep dividing by the factorial of each repeated letter.

5! means "5 factorial" — it's just multiplying all whole numbers from 5 down to 1:

$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$

Why do we do this?

We have 5 letters to arrange. We're asking:

"How many ways can 5 things fill 5 spots?"

• 1st spot → 5 choices
• 2nd spot → 4 remaining
• 3rd spot → 3 remaining
• 4th spot → 2 remaining
• 5th spot → 1 remaining

So total = 5 × 4 × 3 × 2 × 1 = 120

That's all 5! is — the number of ways to arrange 5 distinct things. Then we divide by the repeats afterward.

$120 ÷ 2 ÷ 2 = \boxed{30}$

Even faster mental math:

• 5! = 120
• 2! × 2! = 4
• 120 ÷ 4 = 30

That's it. No formula memorization needed — just divide out the repeats.

The rule in plain English:

Total arrangements = All possible ÷ the repetitions you overcounted