Seven speakers A1, A2, A3,...... A7 were scheduled to speak at a function. In…

Concept: When n distinct objects are arranged in a row and a specific subset of k of those objects must occur in one fixed relative order (not necessarily adjacent), the count of valid arrangements is the total arrangements of all n objects divided by k! — because exactly 1 of the k! possible relative orderings of that subset matches the required order, and this ratio is the same across every arrangement of the remaining objects.

Application:

1. With no restriction, all 7 speakers can be arranged in 7! = 5040 ways.

2. Among themselves, A1, A3 and A5 can occur in 3! = 6 different relative orders, and only the order “A1 before A3 before A5” is allowed.

3. Since each of these 6 relative orders occurs equally often across the 5040 total arrangements, the arrangements satisfying the required order equal 5040 ÷ 6 = 840.

Cross-check: Choose 3 of the 7 speaking slots for A1, A3 and A5 — this can be done in 7C3 = 35 ways. Their order within those chosen slots is forced to be A1, then A3, then A5, so there is exactly 1 way to place them. The remaining 4 slots are filled by the other 4 speakers in 4! = 24 ways, giving 35 × 24 = 840 — the same count.