Find the total number of combinations of the letters a, b, a, b, b (5 letters…

Concept: When a group of items contains repeated types — say p identical items of one kind and q identical items of another kind — the total number of distinct combinations obtainable by choosing one or more items (any number from 0 up to the count available of each kind) equals (p + 1)(q + 1) − 1, where the '−1' removes the single case of selecting nothing at all.

Application: Here there are 2 identical a's and 3 identical b's, so p = 2 and q = 3. The total number of combinations taking some or all of the letters is (2 + 1)(3 + 1) − 1 = 3 × 4 − 1 = 12 − 1 = 11.

Cross-check: this matches a direct count of the combinations possible at each selection size:

1. Choosing 1 letter: a or b — 2 ways.

2. Choosing 2 letters: aa, bb, or ab — 3 ways.

3. Choosing 3 letters: aab, abb, or bbb — 3 ways.

4. Choosing 4 letters: aabb or abbb — 2 ways.

5. Choosing 5 letters: aabbb — 1 way.

Adding these gives 2 + 3 + 3 + 2 + 1 = 11, confirming the total number of combinations is 11.