Consider buying 6 cars from a dealer who has 5 types of cars. In how many ways can this be done if exactly 2 types of cars must be bought?

Question

Consider buying six cars from a dealer who has five types of cars. In how many ways can this be done if exactly two types of cars must be bought?

The answer is given 50. how??

Answer 1

The process of selecting 6 cars out of 5 types with exactly 2 types being selected can be dissected into two smaller choices, Choice 1) pick the two types you’re going to choose from Choice 2) Pick how many you’re going to get of each of the two types you’ve chosen.

the number of ways of making choice 1 is 5 choose 2, or $\frac {5•4}{2}= 10$ because there are 5 options for the choice of the fist type and four choices for the second type, but this counts each pair twice, in two different orders, so we divide by 2.

The number of ways to make choice 2 is 5, because you can have anywhere from 1-5 of the first type of car, but this forces the number you have for the other type of car.

Now the number of ways of making the total choice is just the number of ways of making choice one and the number of ways of making choice 2.

Answer 2

$5C2*5$ First choose the two types and then assign a count for each of the two types. Possible counts are (1,5),(2,4),(3,3),(4,2),(5,1)

Answer 3

There are $\binom52$ possibilities for choosing the $2$ types.

For each possibility for $i=1,2,3,4,5$ we can buy $i$ cars of type 1 and $6-i$ from type 2.

So in total there are $\binom52\times 5=50$ possibilities.