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 2 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?
 2 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?

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lgbasallote
 2 years ago
Best ResponseYou've already chosen the best response.0why is the solution to this 5 * 3? why doesn't it use combination?

blakeadk
 2 years ago
Best ResponseYou've already chosen the best response.0one 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

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1It is orderdependent. Since each shirtpants 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.

lgbasallote
 2 years ago
Best ResponseYou've already chosen the best response.0yes....but why is it 5 * 3 and not combination?

lgbasallote
 2 years ago
Best ResponseYou've already chosen the best response.0the question is the number of ways you can pick...so combination should be used right?

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1We use combinations on orderindependent 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 oneanother."

lgbasallote
 2 years ago
Best ResponseYou've already chosen the best response.0yes...and this doesn't have an order....

lgbasallote
 2 years ago
Best ResponseYou've already chosen the best response.0it's just selection....

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.2Just 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 \[\binom{5}{1}\cdot\binom{3}{1}=5\cdot3=15\]

lgbasallote
 2 years ago
Best ResponseYou've already chosen the best response.0that's another thing i don't get...why multiply? why not add?

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1Well, 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.

lgbasallote
 2 years ago
Best ResponseYou've already chosen the best response.0..but it does answer my question...

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.2The 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.

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1And, 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: \[ S\times P=SP \]Where: \[ S\cap P=\emptyset \]

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1Much 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.

lgbasallote
 2 years ago
Best ResponseYou've already chosen the best response.0actually...it does @KingGeorge ....does that always apply?

lgbasallote
 2 years ago
Best ResponseYou've already chosen the best response.0@LolWolf im not really comfortable with many math symbols....

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1Wait, in my answer or for the book/text?

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.2Sorry, 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.

lgbasallote
 2 years ago
Best ResponseYou've already chosen the best response.0@LolWolf what you just said

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.2First, 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.

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1All 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 \(S\).

lgbasallote
 2 years ago
Best ResponseYou've already chosen the best response.0sorry @LolWolf but i really am not a mathematician....that's too complicated for me....

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.2If you want a slightly larger example, we can count the number of ways we can order a deck of 52 cards using this method.

lgbasallote
 2 years ago
Best ResponseYou've already chosen the best response.0would it turn out as 52!

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.2\[\begin{array}{ccc} \hline \text{Step}&\text{Construction} & \text{Choices} \\\hline \\1&\text{Choose a first card}&52\\ \hline \\2&\text{Choose a second card}&51 \\\hline \\\vdots&\vdots&\vdots\\\hline \\52&\text{Choose a 52nd card}&1\\ \hline \\ \end{array}\] These are all "and then" statements, so you multiply everything together, and you get \(52\cdot51\cdot50\cdot...\cdot2\cdot1=52!\)

lgbasallote
 2 years ago
Best ResponseYou've already chosen the best response.0oh yes,.... i see what you mean by and then now
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