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Use any or all of the methods described in this section to solve each problem. From 10 names on a ballot, 4 will be elected to a political party committee. In how many ways can the committee of 4 be formed if each person will have a different responsibility?

This question snooze out of then local names. Really four. Probably a four B selected Now The important thing here is that off the people selected the rule off, each of them can be different. So because their responsibilities will be different when we make us election in this guest order of selection will be important, because if we select the same two people but in a different order, the responsibilities will be different ends thing selection or the arrangement is actually different, even if the same two people are selecting the different order. So since the selection order is important, this thing we have to apply permutation for this problem. So the number of with in which we can select four people who will be permutation four people out of then no according to permutation formula, this is 10 factorial Aton, then minus four story in which is in fact a real upon six factorial. In fact, you can be return us then James name things. Eight things. Seven. James Dorian Fun spectral yet no weakened against him. Expect a real from the new American denominators. So we get 10 banks nine, which is 90. James Kate things seven and this product gives us fight chosen 40. So there are 5040 different years in we can select this political party committee.

In this problem of basics of content Terry. We can use any method described in destruction to solve each problem. We have given problem from 10 names on a ballot. four will be elected to a political party committee. In how many ways Can the Committee of four B. Form? If each person will have a different responsibility? We have given in names and we have to select four names from it for a political party and this for how different responsibility. So here other matters because here other matters. So we use permutation to select this four member. So we have to compute attain permutation for we know the formula for permutation NPR which is in factorial upon in minus R. Factorial. Using the formula we have 10 factorial upon 10. -46 factorial multiplying for Victoria. Now we expand this 10 factorial using formula and factorial equal to and multiplying weight and -1 factorial. So we have 10 multiplied by nine. Multiplied by eight. Multiplied by seven multiplying with six factorial upon six factorial, multiplying, expanding this four, we have four multiplied by three, multiplied by two, multiplied by one. Now we can divide the six factorial by this. Six factorial. So we have 10 multiply by nine. Multiplied by eight, multiplied by seven whole upon 24. We can divide this 24 with eight. This will be three times, this will be one time and we can divide this night by this three. It will be one time. It will be three times. So we have 10 multiplied by three, multiplied by seven. So it will be for 2 210. So this will be our final answer.

Okay. This question has a situation where we have 10 people. And from there we're signing four of them, two unique committees. So the key thing here is that they're Scient, two unique committees. So it matters who goes to which one. So that should tell us that we're dealing with a permutation here. So let's say we have the four different committees right here. So for the 1st 1 we have 10 choices of who we could seat there. Then for the second, we have nine choices than for third. We have eight in the fourth. We have seven. And that gives us our total number, which is the same thing as the number of permutations from a group of 10. Before signing to four different committees and punching this into our calculator. Here we get 5040 different committee orders.

We have 12 Democrats and 10 Republicans. We need to pick out four Democrats and three Republicans. So how many different ways can this happen? Well, the order in which we picked these people does not matter. That is, the first Democrat chosen doesn't have any power over the 2nd 1 chosen. So there is no order involved in this problem. And when there's no order, you should automatically think of combinations. That is permutations. Order matters combinations. Order does not. So for the Democrats, we have the combination. 12 Choose four that is 12 total. Democrats were picking out four of them, and for the Republicans we have 10 choose three by the same logic, using the multiplication principle, also known as the fundamental principle of counting. If we have a couple of different things that we're doing, and each one happens in a few different ways, we just multiply all the number of ways that this can happen. To get the total options that is, 10 or 12 choose four times 10 choose three gives us the total number of ways that we can select these people, and if you multiply these things together, you get 59,400. So there are 59,400 different ways to select these people