Question

Planing how to car-pool for a trip to Boston in a group of 12 friends (including yourself), you offer to drive your car, where you can take 3 of them.

(a) How many choices do you have if 2 of your friends are feuding and will not sit in the same car together?

(b) How many choices do you have if 2 of your friends will only get together in the car?

Answer 1

(a)

Total number of friends = 12

Since you are driving the car, Total number of friends available for choice = 12 - 1 = 11

Number of persins to be taken = 3

Given: 2 of friends will not sit in the same car.

First calculate the number of ways these 2 friends will be in the same car.

For that, assume both of them as a single friend.

Thus, we have to take 2 friends (1 friend + 1 bunch of 2 frients) from 10 friends (9 frients + 1 bunch of 2 friends).

So,

Number of ways of taking 2 from 10 is given by:

Total number of ways of taking 3 friends from 11 friends is given by:

So,

Number of choices if 2 friends will not sit in the same car = 165 - 45 = 120

So,

Answer is:

120

(b)

Total number of friends = 12

Since you are driving the car, number of friends available for the choice = 12 - 1 = 11

Number of persins to be taken = 3

Given: 2 of friends will only sit in the same car.

The number of ways these 2 friends will be in the same car.

For that, assume both of them as a single friend.

Thus, we have to take 2 friends (1 friend + 1 bunch of 2 frients) from 10 friends (9 frients + 1 bunch of 2 friends).

So,

Number of ways of taking 2 from 10 is given by:

So,

Number of choices if 2 friends will only sit in the same car = 220 - 55 = 45

So,

Answer is:

45