smartass666 wrote:
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?
A. 32
B. 36
C. 48
D. 72
E. 120
There are two different ways to satisfy the given conditions:
Case i: Betty or Veronica sit in the third row
Case ii: Betty and Veronica sit in the 3 middle seats with someone seated between them
Case i: Betty or Veronica sit in the third rowWe can select someone to sit in the third row in
2 ways (either Betty or Veronica will sit here)
We can select someone to sit in the aisle seat in
3 ways (can be Archie, Jerry, or Moose)
There are now 3 people remaining to be seated in the three middle seats in the front row
We can select someone to sit in the leftmost seat in
3 ways (can be any of the 3 remaining people)
We can select someone to sit in the middle seat in
2 ways
We can select someone to sit in the rightmost seat in
1 way
By the Fundamental Counting Principle (FCP), we can complete all 5 steps (and the seat all 5 people) in (
2)(
3)(
3)(
2)(
1) ways (=
36 ways)
Case ii: Betty and Veronica sit in the 3 middle seats with someone seated between themWe can select someone to sit in the aisle seat in
3 ways (can be Archie, Jerry, or Moose)
We can select someone to sit in the leftmost seat in
2 ways (either Betty or Veronica will sit here)
We can select someone to sit in the rightmost seat in
1 way (will be either Betty or Veronica, depending on who is sitting in the left-most seat)
There are now two people remaining to be seated.
We can select someone to sit in the middle seat in
2 ways
We can select someone to sit in the third row in
1 way
By the FCP, we can complete all 5 steps in (
3)(
2)(
1)(
2)(
1) ways (=
12 ways)
TOTAL number of ways to seat all five people =
36 +
12 =
48Answer: C
Cheers,
Brent