## lgbasallote 3 years ago A person has 5 new shirts and 3 new pants. If an outfit consists of a shirt and a pant, how many distinct outfits can the person have?

15

2. lgbasallote

why is the solution to this 5 * 3? why doesn't it use combination?

one pair goes with 5 different shirts to make 5 outfits while another pants pair makes another 5 outfits while the last pants pair makes 5 more outfits which im pretty sure leads to 5+5+5=15 if there is another way you are suppose to do ithe problem i don't know it

4. LolWolf

It is order-dependent. Since each shirt-pants combination is unique, then we have that it is simply the amount of ways in which one can pick any numbering of shirts with any numbering of pants, and have a different combination in each case. I.e. Pants 1, Shirt 2 is a different combination than Pants 2, Shirt 1.

5. lgbasallote

yes....but why is it 5 * 3 and not combination?

6. lgbasallote

the question is the number of ways you can pick...so combination should be used right?

7. LolWolf

We use combinations on order-independent events. E.g. cards. Saying we have a hand of cards 1, 2, 3 is the same as saying we have a hand of 3, 1, 2, as all cards have the same value, it's simply a permutation of the other. Remember that a combination $${n \choose r}$$is an equivalence to: "The number of ways one can pick $$r$$ elements from a set of length $$n$$, such that no two sets of $$r$$ elements are permutations of one-another."

8. lgbasallote

yes...and this doesn't have an order....

9. lgbasallote

it's just selection....

10. KingGeorge

Just to butt in here, but it you can use combinations. You choose 1 shirt from 5, to get $$\binom{5}{1}$$, and 1 pair of pants from 3, to get $$\binom{3}{1}$$. Multiply it together, you get $\binom{5}{1}\cdot\binom{3}{1}=5\cdot3=15$

11. lgbasallote

that's another thing i don't get...why multiply? why not add?

12. LolWolf

Well, yes, it's possible, just not necessary. It can be expressed as an identity of combinations, since we know that any one element will never be a permutation of another unless it's the same, single, element.

13. lgbasallote

..but it does answer my question...

14. KingGeorge

The way I was recently taught to think about it, was by using statements "and then" and "or you could." When you use the "and then" you multiply, and with the "or you could" you add. In this case, you choose a shirt, "and then" choose a pair of pants. However, this doesn't precisely answer your question either.

15. LolWolf

And, we multiply since, if we imagine a set of sets counting each possibility (in this case), the Cartesian product is their total outcome set. E.g. Let's say we have set $$S$$ of shirts and $$P$$ of pants, then, the total set of combinations is: $$S\times P$$, whose cardinality is: $|S\times P|=|S||P|$Where: $S\cap P=\emptyset$

16. LolWolf

Much of this is very well explained an proven in any decent Probability Theory book/textbook. There's a few online from professors, who simply put them up.

17. lgbasallote

actually...it does @KingGeorge ....does that always apply?

18. lgbasallote

@LolWolf im not really comfortable with many math symbols....

19. LolWolf

Wait, in my answer or for the book/text?

20. KingGeorge

Sorry, dealing with some other things now as well. But this can apply to every counting problem I've successfully completed so far. Just a couple weeks ago, one of my teachers recommended a certain method of counting. Give me a minute, and I'll explain it.

21. lgbasallote

@LolWolf what you just said

22. KingGeorge

First, to count something, you're basically counting sets. To count it, we can create a general set with the properties we want, and while we do it, count how many ways to do each step. I'll use this problem as an example. First up, we construct the set. Step 1: Pick a single shirt from a set of 5 shirts. Choices: $$\binom51$$ "And then" Step 2: Pick a single pair of pants from 3 pairs. Choices: $$\binom31$$. Hence, you have $$\binom{5}{1}\binom31$$ ways to dress yourself today.

23. LolWolf

All right, so the "Cartesian product" is when we take an element of each of two sets and concatenate these two to form one more set, which itself becomes an element of our new set, and we do this for every item in each of the two initial sets. Jeez, that's quite a bit of "sets," but it looks more like this: Let's say we have $$S$$ and an element $$a$$ in it, and another set $$T$$ with an element $$b$$ in it, then the set $$\{a,b\}$$ is an element of the Cartesian product, yet, neither $$a$$ nor $$b$$ are, individually. And, sorry for the unnecessary restriction: $$S\cap P=\emptyset$$, I was thinking of something else. As for the other part: the cardinality of a set $$S$$ (its length) is denoted $$|S|$$.

24. lgbasallote

sorry @LolWolf but i really am not a mathematician....that's too complicated for me....

25. KingGeorge

If you want a slightly larger example, we can count the number of ways we can order a deck of 52 cards using this method.

26. lgbasallote

would it turn out as 52!

27. KingGeorge

$\begin{array}{|c|c|c|} \hline \text{Step}&\text{Construction} & \text{Choices} \\\hline \\1&\text{Choose a first card}&52\\ \hline \\2&\text{Choose a second card}&51 \\\hline \\\vdots&\vdots&\vdots\\\hline \\52&\text{Choose a 52nd card}&1\\ \hline \\ \end{array}$ These are all "and then" statements, so you multiply everything together, and you get $$52\cdot51\cdot50\cdot...\cdot2\cdot1=52!$$

28. lgbasallote

wow

29. lgbasallote

oh yes,.... i see what you mean by and then now