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Answer each of the following: labeling your answers clearly Where appropnate; You can leave your answcr terms of factorials or binomial coefficients (in the C(n k) ...

Question

Answer each of the following: labeling your answers clearly Where appropnate; You can leave your answcr terms of factorials or binomial coefficients (in the C(n k) format)A club consists of 12 men and 14 women: In how mnany ways can they choose president; vicc president and treasurer if the president is man and other two officers are women?club consists of 12 men and 14 women: How many dilferent commiltees consisting 0 men and womien Jic possible?Suppose that vou have 20 identical cupcakes t0 di

Answer each of the following: labeling your answers clearly Where appropnate; You can leave your answcr terms of factorials or binomial coefficients (in the C(n k) format) A club consists of 12 men and 14 women: In how mnany ways can they choose president; vicc president and treasurer if the president is man and other two officers are women? club consists of 12 men and 14 women: How many dilferent commiltees consisting 0 men and womien Jic possible? Suppose that vou have 20 identical cupcakes t0 distribute to seven children; In how mny wavs €in this bc done? Suppose that vou have 20 identical cupcakes to distribute to seven children In how many ways can this be done cach child recelves at least one cupcake? e) How many shortest lattice paths are there from (0.0) to (7, 4)? How many shortest lattice paths from (0, 0) to (7,4) Poss through thc point (5, 2}?



Answers

Pizza In an ad for Pizza Hut, Jessica Simpson explains to the Muppets that there are more than 6 million possibilities for their 4forAll Pizza. Griffin Weber and Glenn Weber wrote an article explaining that the number of possibilities is far more than 6 million, as described below. Source: The College
Mathematics Journal.
a. Each pizza can have up to 3 toppings, out of 17 possible choices, or can be one of four specialty pizzas. Calculate the number of different pizzas possible.
b. Out of the total possible pizzas calculated in the first part of this exercise, a 4forAll Pizza consists of four pizzas in a box. Keeping in mind that the four pizzas could all be different, or there could be two or three different pizzas in the box, or all four pizzas could be the same, calculate the total number of 4forAll Pizzas possible.
c. The article considers another way of counting the number in part b. Suppose that only 8 pizzas were available, and they were listed in a row with lines separating each type, as in the following diagram:
$$
\mathrm{A}|\mathrm{B}| \mathrm{C}|\mathrm{D}| \mathrm{E}|\mathrm{F}| \mathrm{G} | \mathrm{H}
$$
A person orders 4 pizzas by placing 4 $\mathrm{X}$ 's in the desired places on the diagram, after which the letters can be ignored. For example, an order for 2 of $\mathrm{A}, 1$ of $\mathrm{C},$ and 1 of G would look like the following diagram.
$$
\mathrm{XX}| | \mathrm{X}| || | \mathrm{x} |
$$
The number of ways this can be done is then the number of ways of arranging 11 objects, 4 of which are X and the other 7 of which are vertical lines, or
$$
C(11,4)=330
$$
Use similar reasoning to verify the answer to part b.

All right, So this man's got three Bonilla, two chocolate, four strawberry, five butterscotch, 14 Children. So 14 factorial on the top. And then our repetitions of three factorial two factorial four and five. Now I'm gonna use the biggest one. Gonna cancel that five out. And so the 14 I'm gonna go all the way down to six. Because from five day on that cancelled out the five factorial in the denominator. Then I have three times. Two times one times, two times one in terms for war comes three times. Two times one. So now I can start canceling additional items out. Ah, let's see, Six times too is 12. So we take the 12 hour four times two is eight, so we can take out the eight, and then the three congest be reduced with the six to make a to Now we can multiply What we have left together 14 times. 13 times, 11 times, 10 times nine Tom. Seven times two. So we're gonna get two million 522,000 520

Okay, So if you're making a committee with two juniors out of seven juniors available and two sophomores out of five sophomores available, what you want to do is find the number of combinations for the juniors and multiply it by the number of combinations for the sophomores. Remember, we learned in the first section of this chapter that when you have something and something in accounting situation, you multiply the two values. So the number of ways to get to Junior's out of seven would be seven shoes, too. Their order doesn't matter. So it's just a combination and the number of ways to get 52 sophomores at if I would be five choose to. So you can compute each of those combinations and seven shoes to gives you 21. And if I choose to give you 10 so when you multiply those together, you have to 10

So for party here. Seven. Factorial. My calculator gets me to 5040. But in the formula we would use the square root of two pi times and which is seven to the seven plus 0.5 times E to the negative seven. And this is equal to approximately 4980.4. To find the year percent or the percent difference, you would do 50 40 minus 4980.4, divided by 5040 times 100 to get percent, and this is approximately equal to 1.18% now, who compete for Part B 10 factorial is equal to 3,628,800. And if you put it into the formula where you can't is covered of two pi times 10 to the 10.5 times E to the negative 10 which is approximately equal to three million 598,695 0.6. So if we do 10 factorial, which is this number divided by the second number here, weren't subtracted and then divided by 10 factorial, which is 3,628,800. Multiply that by a 100%. Will you get that? This is approximately 0.8 three percent.

Okay, The question is a man aboard three vanilla ice creams, three vanilla ice cream, corn's two chocolate ice cream cones and whole Starbury ice cream cones. Life but your squash ice cream cold. Now he's going to distribute all these ice cream cones in 14 Children. So in how many of these can he distributed Now, first of all, we have to look that considered an example. We have to this kind of marbles into this kind of marbles. And if we do the arrangement of this considered this the arrangement off four marbles. And if I exchange this marble with this as their scene, so the arrangement will remain the same. We're going to use this trick. No, is there are 14 different ice creams, so if they were all different permutation or the number of these in which he could have arranged it or given to the Children would before would have been 14 15. But if he exchanged one thing, they love the other one. The regiment remain saying, so we have to exclude all those arrangements in which we're exchanging one. Then they love flavored corn with the other than enough lyrical and same likewise we're going toe exclude or those permutations. And this were exchanging one chocolate cone with the other chocolate kun and like ways. So the answer is two. It's life. Double to five to legal. This is the formula off and one and two and three. And so, for example, then the sum of all these and one plus and two plus n three less sinful. But you're you divided by any one pictorial and two pictorial and three pictorial and full effect on you when we have four different kinds of objects and one kind is like and one in number. The other kind is into a number that the oneness and three no booed off number of objects in that kind. Then this is the formula for mutation. This is the one that we have applied.


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