How many ways can the trip be outfitted?

Question

Assume that the camper has 3 jackets, 7 sleeping bags, and 12 different packages of food. An outfit consists of 2 jackets, 2 sleeping bags, and 9 packets of food.

How many ways can the trip be outfitted? 

Answer 1

When you are picking say 2 jackets out of 3, that is a "combination" because, it doesn't matter what order you pick the jackets in, the two jackets are still the two jackets (doesn't matter if one was picked first).

So the possibilities for jackets is ₃C₂ and this is typically something you can type in your calculator to determine. If not, you have to use the equation:

It turns out that ₃C₂ = 3. You can easily see this if you just label the jackets as A, B, C. The possible choices are AB, AC, and BC.

The number of possibilities for sleeping bags is ₇C₂ and the number of possibilities for packets of food is ₁₂C₉

Once you get each of those values, just multiply them together to get the total number of possibilities. So:

Total # of ways the trip can be outfitted = (₃C₂)•(₇C₂)•(₁₂C₉)

Answer 2

Let's first consider how many ways she can bring 2 jackets. She is choosing 2 jackets from 3, so we use the formula for 'n choose k', where n=3 and k=2 (see this website for the formula: https://www.hackmath.net/en/calculator/n-choose-k). This gives us 3.

For the sleeping bags, we are choosing 2 from 7 (n=7, k=2), which produces 21.

For the food, we choose 9 from 12 (n=12, k=9), which produces 220.

We multiply all 3 numbers together to get the *total* number of ways the trip can be outfitted; this gives us 13860.