Find the number of ways in which one or more letters can be selected from the…

Concept: When selecting items from a multiset (a collection where items can repeat), each distinct item that occurs n times offers (n+1) choices -- you may include 0, 1, 2, ..., or all n copies of it in a selection. Multiplying the choice-counts of all distinct items together counts every possible selection, including the empty selection (choosing nothing); subtracting 1 removes that empty case, giving the number of ways to select one or more items.

Application: The word AAAABBCCCDEF has 4 A's, 2 B's, 3 C's, and 1 each of D, E, F.

1. Ways to choose how many A's to include: 4 + 1 = 5 (0 through 4 A's)

2. Ways to choose how many B's to include: 2 + 1 = 3

3. Ways to choose how many C's to include: 3 + 1 = 4

4. Ways to choose how many D's to include: 1 + 1 = 2

5. Ways to choose how many E's to include: 1 + 1 = 2

6. Ways to choose how many F's to include: 1 + 1 = 2

7. Multiply all choice-counts together: 5 x 3 x 4 x 2 x 2 x 2 = 480 total selections, including the empty one

8. Subtract the single empty selection (choosing no letters at all): 480 - 1 = 479

Cross-check: Re-multiplying step by step confirms the arithmetic -- 5 x 3 = 15, 15 x 4 = 60, 60 x 2 = 120, 120 x 2 = 240, 240 x 2 = 480, and 480 - 1 = 479 -- so the count of ways to select one or more letters is confirmed independently.

Therefore, the number of ways to select one or more letters from AAAABBCCCDEF is 479.