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Counting PracticeQuestion. If 5 people are asked their birth months and the results written down one at time, 12^5 different sequences might be created. In how many...

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Counting PracticeQuestion. If 5 people are asked their birth months and the results written down one at time, 12^5 different sequences might be created. In how many of these is there a repeated month?Answer:ENTER[f you are finished trying this question; click on "Next" to proceed.Next

Counting Practice Question. If 5 people are asked their birth months and the results written down one at time, 12^5 different sequences might be created. In how many of these is there a repeated month? Answer: ENTER [f you are finished trying this question; click on "Next" to proceed. Next



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Birthday Problem In how many ways can 5 people all have different birthdays? Assume that there are 365 days in a year.

Suppose we've got five different people A, B, C, D and E. Each one of them has to have a different birthday from the others. So how many different ways can this occur? Well, let's consider. So for person A. Let's assign him whatever birthday he wants. That is. He has 365 different options for his birthday, so that is 365 days in a year, so you can pick whatever birthday now. Person B has almost that many options. However. They don't get the choice of whatever day person a pit, so they only have 364 options. Now, Person See can't pick person A's Day or Person B's Day. They only have 363 Person D. Similarly, only has 3 62 and Person E has 3 61 So those are the different options that each person has to pick their birthday. Assuming this is in a perfect world where we could pick our birthdays instead of just being assigned them well, now we can determine how many different ways this could happen. That is, we could have January 1st through fifth be there birthdays, or they could be scattered out one in January 1 in April 1 in June. Whatever so energy to this we invoke the fundamental principle of counting what the fundamental principle of counting says is if we have these different things, each of which has some different options, we multiply the number of options together to get the total number of ways that this can occur. That is 3 65 times 3 64 times 3 63 times 3 62 times 3 61 And the final result of this multiplication is an incredibly large number. So I won't make you do it by hand. We have, and this will take a second to write. We have this phenomenally large number that is six trillion 302,555,018,760 different ways this can happen. That is a lot of different ways for this to occur more than the number of people that are in the world now. One other thing that's worth noting with this is that this is actually the permutation of 365 and five. That is, if you per mute three or 65 the number of days in a year by five different people. You will also get this large number which is 3 65 multiplied down through 3 61 Either way, you will get this number as the number of different ways that people can have different birthdays.

Suppose we've got five different people now. All we know about them is that they were not born on the same day. That is, they have different birthdays. So how many different ways can this occur? Well, there's 365 total days in the year, and we need to choose out five different days. For each of these five people, the order doesn't really matter. We don't need to have that. This guy's birthday was first than his than his. We could have them be in any order. So since order doesn't matter, this means combination. Whenever you hear that order doesn't matter. You should generally think of combinations. So we're going to be doing a combination of 3 65 truce. Well, choose five, because there are five people who have birthdays. So we need to pick out five random days throughout the year, which are not the same day to make their birthdays. So if we do the combination 365 choose five, we get an extremely large number. I'm not going to be using the formula from the book, but rather just a calculator, because this number is enormous. The number of combinations or rather, the number of possible birthdays. Sets of birthdays is 52,000,000,521 million 291,823. So that's the number of ways that these five people can have birthdays, which are all on different days, and that is equal to 3 65 choose five.

So in how many ways can five people have different birthdays? Yeah, and we can answer this a few different ways, but I think this way works really, really nicely to put down the five people. Those are your five people in whatever order you have them. This person can have any of the 365 birthdays now, the second person cannot have the same. So you're restricted on 360 for this person, can't have two of these birthdays that have already been used up. So 363, and you get the pattern 362 and the last person has 361. This is also equivalent to a permutation Of 365, choose five. And if we do that, 365. Yeah, let's quit. Just do this calculation. And I'm almost done. We get a huge number. We get approximately 6.30255 five times 10 to the 12th power. So it's a very large uh number of ways that you can have the 3 65 you know, five people have different birthdays and you would expect

For this problem, we are asked how many six letter sequences are possible that use the letters, F, F, A A and F F. So our first step here, step one will be to place the A's. So I'll note that we are trying to place this into six boxes, so we have six choices. But we do need to note that we can't distinguish the two A's from each other, so we don't have six times five choices. Instead, we have the number of combinations of six items taken two at a time, number of choices, and that is going to be 15. We have 15 different possible ways to place the A's at our first step And then our 2nd step. Well, that is to place the efs. But no matter how we arrange the efs, the sort of state that we're in has already been decided based on where the A's are. So there's only one real meaningful way to arrange those efs. So we have one choice at this point. So the number is just going to equal 15


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