How many ways can Laura wear her favorite clothes without repeating the same exact outfit?
So this question asks how many outfits (consisting of one item of each of the five groups) can we make without repeating. To best explain this question, lets simplify it for a moment and say that Laura wants to know the combinations of only the 3 jackets, 2 shoes, and belt she can wear without repeating.
The three jackets will be represented as A1, A2, and A3,
The two shoes will be represented as B1 and B2,
and the Belt will be C1
Since we can only have one from each group, these are the only combinations possible:
A1 B1 C1
A1 B2 C1
A2 B1 C1
A2 B2 C1
A3 B1 C1
A3 B2 C1
So there are six possible combinations of just these three items. Mathematically, we could express this as 3 X 2 X 1 which is also six.
Another way of thinking about this is that we have three articles of clothing to select to start. Once we select a jacket, shoes, or belt we have two articles of clothing left to select. Once we select one of those, we have one article of clothing left to pick. So to find the number of combinations can we can multiply n by n-1 until we reach 0 or 3 X 2 X 1.
In the math world this can be represented as a factorial or 3! which is an easy way to represent 3 X 2 X 1.
We can do this for any number (n) articles of clothing. So let's go back to our original problem:
Laura needs to select outfits from 5 groups of clothing. It doesn't matter which order she picks the clothes in so the number of combinations would be 5! since n=5. If you listed every possibility out like I did above, you would indeed get the same answer as 5 X 4 X 3 X 2 X 1. If n=2 then you would do 2! or 2 X 1 which is 2. That would be asking now many combinations of the two shoes and belt can be made without repeating or (B1 C1) AND (B2 C1).
1! = 1 (Only 1 Possible Combo of Belts) = 1
2! = 2 (Only 2 Possible Combos of Shoes and Belts) = 2 X 1
3! = 6 (6 Combos of Jackets, Shoes, and Belts) = 3 X 2 X 1
4! = 24 (Number of Combos of Skirts, Jackets, Shoes, and Belts) = 4 X 3 X 2 X 1
So what is 5! or the number of combs of Sweaters, Skirts, Jackets, Shoes, and Belts?
Hope this helps!