Problem Solving

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

OA is C

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?
32
36
48
72
120

Good = Total - Bad.

Total = arrangements with Archie, Jerry or Moose in the aisle seat:
Number of options for the aisle seat = 3. (Archie, Jughead, or Moose)
Number of ways to arrange the 4 other people = 4*3*2*1.
To combine these options, we multiply:
3*4*3*2 = 72.

Bad = arrangements with Archie, Jerry or Moose in the aisle seat BUT with Betty next to Veronica:
Number of options for the aisle seat = 3. (Archie, Jughead, Moose).
Number of options for the third row seat = 2. (Anyone but Betty and Veronica, since in a bad arrangement they sit next to each other.)
Number of options for the middle of the 3 remaining seats = 2. (Must be Betty or Veronica so that they sit next to each other).
Number of ways to arrange the 2 remaining people = 2*1.
To combine these options, we multiply:
3*2*2*2 = 24.

Good arrangements = 72-24 = 48.

The correct answer is C.

Video Solution



-Patrick

Ignoring the aisle seat for the time-being, here are the seats that are left:

Triple and single
000 and 0

Here are all the possibilities that keep B and V separate (with B to the left of V):

B0V 0 x 2 arrangements of the other ("0") sitting arrangements
00B V x 2
0B0 V x 2
B00 V x 2

Which totals 2 x 4 = 8 combinations

Similarly there are 8 more combinations with B an V reversed (i.e. V sitting to the left of B)

This gives us a total of 8 + 8 = 16 combinations.

However if we consider that one of A, J or M sits in an aisle seat, this gives us 16 x 3 combinations possible

So, the answer is 16 x 3 = 48 [ANSWER C]