How many 3-letter selections can be made from COLLIDER

Key takeaway: This is a combinations question, but the repeated letter changes the count. The main thing is to decide whether you are selecting letters or forming arrangements, because that changes the method completely.

The word COLLIDER has 8 letters:

$C, O, L, L, I, D, E, R$

So there are two Ls and the other letters are distinct.

(i) Exactly one L

We need 3 letters in total, with exactly one L.

• choose 1 L from the 2 available: $\binom{2}{1}=2$
• choose 2 more letters from the 6 non-L letters: $\binom{6}{2}=15$

So the number of selections is

$2\times 15=30$

(ii) No restrictions

Now choose any 3 letters from the 8 available letters, treating the two Ls as separate letters in the word.

$\binom{8}{3}=56$

So the answer is 56.

---

Pitfalls the pros know 👇 Do not treat the two Ls as the same object when the question is counting selections from the letters in the word. That would lose valid cases.

Also, this is a selection question, not a permutation question. If the order mattered, the answer would be different.

What if the problem changes? If the question had asked for exactly two Ls, the count would be

$\binom{2}{2}\binom{6}{1}=6$

If it had asked for 3-letter arrangements, then you would count different orders separately, so the method would switch from combinations to permutations.

Tags: combinations, selection, repeated letters