A group of 5 friends—Archie, Betty, Jerry, Moose, and Veronica—arrived at the movie theater to see a movie. Because they arrived late, their only seating option consists of 3 middle seats in the front row, an aisle seat in the front row, and an adjoining seat in the third row. If Archie, Jerry, or Moose must sit in the aisle seat while Betty and Veronica refuse to sit next to each other, how many possible seating arrangements are there?

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120

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Official Explanation

Official

Conditions:

Archie, Jerry, or Moose must sit in the aisle seat (positive condition).

Betty and Veronica refuse to sit next to each other (negative condition).

The positive condition must be fulfilled first.

Seat layout: (aisle seat) * (three front row middle seats) * (third row seat).

Step 1: Choose someone for the aisle seat.

Only Archie, Jerry, or Moose can sit here:

3

options.

Step 2: Fill the three middle seats and the third row seat with the remaining 4 people, ensuring Betty and Veronica are not adjacent in the middle seats.

Suppose Archie takes the aisle seat (the count will be the same for Jerry or Moose).

Remaining people: Betty, Jerry, Moose, Veronica.

We have 4 seats to fill: three adjacent middle seats (Seats 1, 2, 3) and one third-row seat.…

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What this question tests

What it tests

Whether you can count arrangements and selections correctly — permutations vs. combinations.

Common trap

Overcounting: treating a selection as ordered when order does not matter, or forgetting to divide by duplicates.