Question
How many 4-letter words (with or without meaning) can be formed using the letters of the word “EQUATION” if no letter is repeated?
Solution — Step by Step
EQUATION has 8 distinct letters: E, Q, U, A, T, I, O, N.
We need to choose 4 letters from 8 and arrange them — that’s .
Final answer: words.
Why This Works
A “word” here means an ordered arrangement of letters. So this is a permutation, not a combination. The first slot can be filled by any of 8 letters, the second by any of the remaining 7, and so on — the multiplication principle gives .
If the question had asked about choosing 4 letters (set selection, no order), it would be .
Alternative Method
Fill slots one by one:
- Slot 1: 8 choices
- Slot 2: 7 choices (one letter used)
- Slot 3: 6 choices
- Slot 4: 5 choices
- Total: .
This is the multiplication principle in raw form, identical to the formula.
Permutation = “arrangement” (order matters). Combination = “selection” (order doesn’t matter). When the question uses words like “arrangements,” “words,” “queue,” or “order,” it’s a permutation. When it uses “team,” “committee,” or “selection,” it’s a combination.
Common Mistake
Confusing “no letter repeated” with “use each letter once” — they’re the same here because we have 8 distinct letters. But if the source word had repeated letters (like “BANANA”), we’d have to deal with identical letters, which is a different formula.