In how many ways can the letters of the word OMEGA be arranged so that O and A…

Concept: When arranging n distinct items in a row and some items are restricted to specific positions, first arrange the restricted items in those positions, then arrange the remaining items in the remaining positions. By the multiplication principle, the total number of arrangements equals the product of the arrangement counts of each group.

1. OMEGA has 5 distinct letters: O, M, E, G, A.

2. The two end positions must be filled only by O and A. These two letters can be arranged between the two ends in 2! = 2 ways (O...A or A...O).

3. The remaining 3 letters (M, E, G) fill the 3 middle positions, which can be arranged among themselves in 3! = 6 ways.

4. By the multiplication principle, total arrangements = 2! x 3! = 2 x 6 = 12.

Cross-check: without any restriction, 5 distinct letters can be arranged in 5! = 120 ways. There are 5 x 4 = 20 ways to choose an ordered pair of letters for the two end positions, and only 2 of these 20 ordered pairs use exactly O and A (either order). For each such valid end-pair, the remaining 3 letters can be arranged in 3! = 6 ways, giving 2 x 6 = 12 valid arrangements - confirming the result above.