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Determine whether each situation involves a permutation or a combination. Then find the number of possibilities.How many ways can a hand of five cards consisting of...

Question

Determine whether each situation involves a permutation or a combination. Then find the number of possibilities.How many ways can a hand of five cards consisting of four cards from one suit and one card from another suit be drawn from a standard deck of cards?

Determine whether each situation involves a permutation or a combination. Then find the number of possibilities. How many ways can a hand of five cards consisting of four cards from one suit and one card from another suit be drawn from a standard deck of cards?



Answers

Determine whether each situation involves a permutation or a combination. Then find the number of possibilities.
How many ways can a hand of five cards consisting of four cards from one suit and one card from another suit be drawn from a standard deck of cards?

Okay, This question is gonna ask us about the number of different ways we could get certain poker hands. And really, we just need to know that a poker hand is formed from any five cards in a normal day. So the 1st 1 asks, How many ways could we get a poker hand with four queens in it? So this is equal to the number of ways to get the Queen's times the number of ways to get everything else. So with the four queens, there are four queens in the deck one pursuit. So four queens and we're picking four of them times the choices for the last card. So there's 52 cards in a deck, which means that for the non queens, we have 48 to pick from, and we're choosing one since is just 48 then we want no face card, so we need to figure out the number of face cards so we know how many we can avoid. So for the face cards, we have the Jack, a queen and a king, and each of these therefore one pursuit because we have four jacks for queens and four cakes. So this means that we have 40 cards to pick from and we want all five to come from there. So our answers just 40. Choose five because we don't want any of the 12 face cards and plugging the send 40 choose five is 658,000 at eight Done. It wants exactly to face cards. So we have a number of ways to pick the face card times the number of ways to pick the non face cards. So we said over here that they're 12 total face cards. We want two of them times the remaining 40 cards where we're gonna pick three of them. And if we do this, we have 12. Choose to Times 40 choose three, which gives us an answer of 652,000 and a different ways that that scenario could play out. Then, For Part D, we have at least two face cards, so this is equal to the number with two plus the number with three plus the number with four. Close the number of five. So for the 1st 1 with two, we have 12 face cards picking too times 40 non face picking three plus 12 face cards picking three times 40 non face picking, too, plus 12 face cards aching four times 40 picking one plus 12 face cards. Picking five times 40 picking zero. And if we do this one, this requires a lot of computation, but we get 844,000 272 after we punch all those numbers in. And something to note with these types of problems is that notice that the top brackets add up to 52 all the time and the bottom bat brackets add up to five every time. And that's no coincidence. Then, for Part E, it says one heart, two diamonds in two clubs. So we have the number of ways to pick the heart, which there are 13 hearts because we have a 52 card deck divided equally amongst four suits. So have 13 pursuit. So a 13 choose one times 13 shoes to for the diamonds times 13 shoes to for the clubs. And if we do that, that's just 13 shoes too squared. Times 13 which gives us 79,092 and that's our final answer

Okay, So the number of ways that you could have five cards and a hand where three cards were from one suit and two cards or from another suit, you really have to break this down quite a bit. Those three cards from one suit those could be three hearts or three diamonds or three states or three clubs. And then when we get all those numbers in that whole part figured out, we're gonna multiply by because of the. And remember in counting when it's Andy multiplied by the never ways to get two cards from another suit. Well, if you've already taken three cards from one suit, you only have three more suits to choose from. So it could be three. Excuse me. Could be two cards from some suit that you haven't yet used. I don't know what it is or two cards from another suit you haven't yet used, whichever that is, or two cards from that other suit you haven't yet used. So there's only three options there because you already used once you, you only have three suits left. Now let's work the combinations in, so there are 13 hearts available. And if you're going to choose three. That would be 13. Choose three for or when we can't, we add, There are 13 diamonds available. You're choosing three. So this is 13. Choose three also, plus there are 13 spades. You're choosing three, so this is also 13. Choose three, and for clubs, it would also be 13 to 3. So that whole thing, added together, is going to be multiplied by the next part. So, uh, pick one of the suits that's left. So one of the suits that's left has 13 cards left in it, so that would be 13. Choose to. There's another suit that's left. It also has 13 cards left in it. So that would also be 13 shoes, too. And they're still yet one more soup that's left and there's still 13 cards in it. So that would be 13 shoes, too. So one or the other or the other. That's why we're adding okay, so if we simplify the left part, we have four times 13. Choose three, and then when we simplify the right part, we have three times 13 choose to, and when we find those numbers, we get four times to 86 times. Three time 78 Multiply all that together and you get 267,000 696.

The game. Booker. There we have the gold standard cars. 52 cards on a no, in a handoff. Booker high cars are drawn randomly, so we have to find out a probability off. Well, getting ah, or the five guards from drawn in one hand. Uh, off seen suit. So in a decor to tow cars. Uh, we have we have, uh, four different soups. Like, ah, eat suit Contains 13 guards each and the soups out off us hard. Oh, our diamond club and speed. So we have to flying the probability that in one of the five guards drawn ah, out of scenes seems to So we have 52 cards and five guards are drawn. Five guards, air drawn, the samples Piece off. Uh uh. That you can't and propose ample space. Ah, number the mentioned examples. Biss, uh, call menacing off you too. Come on. Five. So NFS comes out to be, which is by nine, eight, nine, six. So five guards off any type getting it wrong from 52 cards in these many number please. Now our even Oh, uh, getting even days five guards seems to So this can be a calculator the number off for ways in which this can. Britain ah number of elements in the U Grant can be calculator in this year since there are four suits, so he a given so it can be any problems for tapes observes. So the number of ways it can brittleness see for number one these are in different in different, uh D period number off raise five Guards can be drawn from the given soup so of the skin very or reminiscent of 13 Boma play. This is nothing but for more depend bay. See Dean go a place so the probability off e occurring better be number of elements off E upon number of elements off sample space This this will be acquittal More fame's, uh, communists in a factory in coma place of born, uh International, The pickle Coma. This is a finance form of the answer. So opera solving this get a clue Cedo born I see one 90. This is a very of the probability we get this given groups

So we have 52 cards and we want hands of five. Yeah. So we have 52 factorial over 50 to minus five, which is 47 factorial and then an additional five. So what does that give us? What? We take out the 47? We have 52 times. 51 times 50. Just 49 times 48 over five times or times three times, two times one. And so now we need to either multiply top and bottom and then divide. Or if we can do some simplifying, weaken. Do that. Um, so five times two is 10. And so if I take that out, I have five There. 48 can divide by four to get 12 and 51. Condemn vied by three to get 17. And so that's gonna give me 52 times. 17 times. Five times. 49 times 12. Which gives me two million 598,000 960


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