To find the number of ways to select 3 letters from the set $\{A, A, B, B, B, C\}$, we need to account for cases where letters are repeated. Here's the step-by-step breakdown:

Step 1: Possible combinations of letters

We analyze based on the possible compositions of the chosen letters:

1. Three distinct letters (A, B, C):

   • The selection is $\{A, B, C\}$.
   • There's only 1 way to select this.

2. Two of one letter and one of another:

   • Two A's and one B: $\{A, A, B\}$, which is valid since we have 2 A's and 3 B's.
   • Two A's and one C: $\{A, A, C\}$, valid as we have 2 A's and 1 C.
   • Two B's and one A: $\{B, B, A\}$, valid as we have 3 B's and 2 A's.
   • Two B's and one C: $\{B, B, C\}$, valid as we have 3 B's and 1 C.
   • Total for this case: 4 ways.

3. Three of the same letter:

   • Three B's: $\{B, B, B\}$, which is valid as we have 3 B's.
   • No other letter appears three times.
   • Total for this case: 1 way.

Step 2: Total number of ways

Adding up all the cases: $1 + 4 + 1 = 6 \text{ ways.}$

Final Answer:

There are 6 ways to select 3 letters from $\{A, A, B, B, B, C\}$.

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Would you like more details on the logic or any specific step?

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Related Questions:

1. How does repetition in elements affect combinations?
2. How many ways can 3 letters be selected if all were unique (no repetition)?
3. How would the result change if there were 4 B's instead of 3?
4. Can you calculate the combinations using binomial coefficients for validation?
5. What if the selection order mattered? How many permutations would be possible?

Tip:

Always consider the impact of identical elements on combinatorial calculations—this ensures you don't overcount or undercount.