Three bands and two comics are performing for a student
talent show. How many different programs (in terms of order)
can be arranged?
5 acts, permute all 5. That's 5P5 = 5! = 120 ways.
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How many if the comics must perform between bands?
Choose the band to perform first in 3 ways.
For each of those 3 choices for the 1st act, there are
2 ways to choose the comic for the 2nd.
So that's 3×2 or 6 ways to choose the first 2 acts.
For each of those 3×2 or 6 ways to choose the band to go
first and the comic to go second, there remain 2 bands to
choose to go 3rd.
So that's 3×2×2 or 6×3 or 18 ways to choose the first 3
acts.
For each of those 3×2×2 or 12 ways to choose the 1st 3
acts, there remains only 1 comic to choose to go 4th.
So that's 3×2×2×1 or 18×1 or 18 ways to choose the first 4
acts.
Finally there is only 1 way to choose the last act, which
is the only remaining band.
Answer: 3×2×2×1×1 or 12 ways.
Checking by writing them out:
1. Band A, Comic 1, Band B, Comic 2, Band C.
2. Band A, Comic 2, Band B, Comic 1, Band C.
3. Band A, Comic 1, Band C, Comic 2, Band B.
4. Band A, Comic 2, Band C, Comic 1, Band B.
5. Band B, Comic 1, Band A, Comic 2, Band C.
6. Band B, Comic 2, Band A, Comic 1, Band C.
7. Band B, Comic 1, Band C, Comic 2, Band A.
8. Band B, Comic 2, Band C, Comic 1, Band A.
9. Band C, Comic 1, Band A, Comic 2, Band B.
10. Band C, Comic 2, Band A, Comic 1, Band B.
11. Band C, Comic 1, Band B, Comic 2, Band A.
12. Band C, Comic 2, Band B, Comic 1, Band A.
Edwin