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What is the order of each of the following permutations?a. (124)(357)b. $(124)(3567)$c. $(124)(35)$d. (124)(357869)e. $(1235)(24567)$f. $(345)(245)$...

Question

What is the order of each of the following permutations?a. (124)(357)b. $(124)(3567)$c. $(124)(35)$d. (124)(357869)e. $(1235)(24567)$f. $(345)(245)$

What is the order of each of the following permutations? a. (124)(357) b. $(124)(3567)$ c. $(124)(35)$ d. (124)(357869) e. $(1235)(24567)$ f. $(345)(245)$



Answers

Write
all
permutations
of
the
letters $ A $, $ B $, $ C $,
and $ D $.

For this question were supposed to find the number that number of distinguishable permutations for the letters A, B, C B B B, you e. As you can see, the total number of letters here escaped that That means the routing number of permutations possible escape factorial. Not since d here has been repeated trace We know that it increases the number of permutations by a factor off three factorial does the white eight factorial by three factorial. You can also see that e is repeated twice. This increases the total number of formulations by a factor of two factorial. Thus we did we divide the factorial biting factorial this world. So we have eight factorial divided by three factorial into two factorial. If you simplify this, this comes to 3360 and this is the total number off distinguishable permutations possible

All right, so this time we have Hey, you're revealing is not working e a b b b sensing sick. And so we want to know how many different ways we can put those letters. But because they're repetitions were using the repetition formula. There are nine of them. So that's nine factorial on the top, and then we have three A's. So that's three factorial. We have three b, so that's three factorial and we have three C's. Now Calculator can do that really quickly, or I'm gonna write that out so you can see what that looks like. These numbers are small enough. I can do that. So non factorial means just that and it on the bottom. We have three factorial and we have it three times, so we're gonna have some things that will cancel. For example, this will cross out three times two is six so we can take that one away, and then we can start simplifying. So four will cancel out That too, for one, of course, is out. And the nine here will cancel out this three. And so now we can multiply those numbers across three times eight times seven times five times two, and we're going to get a total then of 1680 possibilities

All right, so we have a B c d. So we're gonna choose five objects. So that's five factorial on top. We have one that repeats itself. So that's two factorial. And then we have 31 so they really don't. You can put a man if you want, but they're not gonna change anything, cause your small claim I want. So we're not gonna do that. So what does that mean? That means we have five times, four times, three times, two times one over to Tom's one, which those of course, will cancel out, and we're left with 60.

All right. So our list, this Tom has a B c three days and to ease so notice that we have a total of eight letters to pick from. So that means that eight factorial is on the top, and then we have some repetitions. We have three ds and to ease, so we're gonna have a three factorial on a two factorial down there so we can write that out so that you can see how that cancels out. Eight Factorial is on the top, and that just means eight. And then you descending down. Then we have three factorial into factorial and so you can cancel out. Well, three times two is six. So I can take the six away and then two and one I can take out the two in the one, and I could multiply together what is left. And so that's gonna be eight times. Seven times five times four times three for a total of 3000. 360


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