My question is:
A rental car agency has 12 identical cars available and 7 identical vans
a) If the group needs to rent four cars and two vans, in how many different ways can they select their vehicles? b) A group taking a field trip needs to rent six vehicles. In how many different ways is this possible?
For (b) I get an answer of 27,132. But for (a) I get 537.
I know the answer to (a) is 10,395 but am unsure how to get there.
What am I missing here?
On
The order of selection does not matter. The number of ways to select four of the twelve cars is $\binom{12}{4}$. For each such selection of the four cars, the number of ways of selecting two of the seven vans is $\binom{7}{2}$. Hence, the number of different ways the group can select their vehicles is
$$\binom{12}{4}\binom{7}{2} = 10 395$$
The answer to a) is
$${12\choose 4}\times{7\choose 2}=10395$$ Out of the $12$ cars, we must pick $4$ of them, which gives us ${12\choose 4}$ options. Out of the $7$ vans, we must pick $2$, which gives ${7\choose 2}$options.