A man has 3 pairs of stacks, 9 shirts, and 7 ties that all coordinate. How many different outfits of slacks, shirts, and ties can he put together?

Answer 1

To answer this type of question, it's a fairly simply math operation - but I want to talk through the intuition so when this type of problem comes up again, you'll have the tools to calculate it quickly.

Imagine we pick the first pair of slacks (let's call them the khaki slacks) and the first shirt (let's say it's a sky blue shirt). How many combinations can we make if we pick that shirt and those slacks? Well, the only thing left that we could pick from are the 7 ties - in other words, we have 7 possible outfits if we've already chosen a shirt and a pair of slacks. We can calculate that by looking at the number of possible choices for each of the three clothing pieces.

1 Shirt (in this example, we picked sky blue)

1 Pair of Slacks (khaki)

7 Ties

The calculation is 1 x 1 x 7 = 7. That makes sense, because we know what shirt and pants we're wearing, but we could pick from 7 different ties.

Now, imagine we suddenly weren't sure if we wanted to wear those khakis anymore. In this example, we still want to wear the sky blue shirt, but now we're not sure about the tie or the slacks. So how many options do we have?

Well, we know that if we were to go back and pick the khakis that there are 7 possibly combinations with blue shirt, khaki pants, and 7 ties. Well, the number of possibilities is the same if we pick a different pair of slacks; let's say we have a gray pair of slacks. Thus, we'd have a blue shirt (1 possible option), gray pants (1 possible option), and the 7 ties (7 possible options). The calculation is again 1 x 1 x 7 = 7 possible combinations if you pick a gray pair of slacks and a blue shirt. Notice how no matter which pair of slacks we pick, we'd have 7 total options after picking that pair of slacks. And since we have 3 pairs of slacks... we can just add the total combinations. 7 + 7 + 7 = 21 possible combinations.

So now we consider that we had 3 options for slacks in the first place - and we can just use the same math equation as before. Total possible shirts x Total possible slacks x Total possible ties = total combinations. If we're still set on a single blue shirt, then that's 1 possible shirt. We have 3 possible slacks, though, and 7 possible ties. 1 x 3 x 7 = 21 (the same number we got doing the math the longer way).

Well, if we change our mind about the shirt, we have the same issue - now we have to re-examine how many options we have. But remember, when we picked a blue shirt, we know that we had 21 total options. The same logic applies from the pants section - no matter which shirt we pick, we'd have 21 options left after we selected that shirt. Since we have 9 different shirts, we could add 21 + 21... (9 times), but that's the same as saying 21 * 9, right? So if you do that math, you get 189.

That's the same as using the equation we talked about earlier: possible shirts * possible pants * possible ties = total combinations. So possible shirts (9) * possible pants (3) * possible ties (3) = 189.

Long story short: multiply each number of options by the other numbers of options and you'll get the anwer.