pizza: 8)A pizza place offers 6 different cheeses and 12 different toppings. In how many ways can a pizza be made with 1 cheese and 3 toppings? 9)How many ways can you purchase 5 CDs if there are 6 to choose from, 2 cassettes if there are 5 to choose from, and 4 DVDs if there are 8 to choose from? Someone please help, I'm having a really hard time trying to figure these out.|||8. Lets deal with the toppings first! You need COMBINATIONS three from 12. The first choice of topping can be any one of 12 The second choice can be any one of 11 (one topping has already been chosen) The third choice can be any one of 10 (2 toppings have already been chosen) Therefore there are 12x11x10 = 1320 PERMUTATIONS of getting 3 toppings. HOWEVER - think about it! A peperoni, mushroom and onion pizza is exactly the same pizza as a mushroom onion and peperoni pizza. Both pizzas have exactly the same toppings but were just seleted in a different order. In fact these three toppings can make exactly the same pizza in 3! or 6 different ways. (PMO = POM = MOP= MPO=OPM=OMP) For each and every combination of three toppings there will always be 6 identical pizzas. Therefore we need to divide the 1320 permutations described above by 6 to allow for this factor. Therefore there are 1320 / 6 = 220 different combinations of toppings. Now the cheeses there are 6 different choices, Each one of these choices can go with the 220 toppings choices. Therefore the answer to your question is 6 cheese choices x 220 toppings combinations = 1320 different pizzas. In maths this would be given by the formula (6! / 5! 1! ) x (12! / 9!3!) 9. The first part is easy it is 6! / 1! 5! There are 6 DVDs and you have to select 5. Therefore there are 6 ways that one can be left out of the selection. Technically it is Combinations 5 out of 6. The first can be any of 6, the second can be any of 5, the third can be any of 4 etc etc 6x5x4x3x2 = 720 Permutations Then like the pizza toppings the 5 selected can be arranged in 5! different ways the first any of 5, the second any of 4 the third any of 3 etc etc therefore there are 5x4x3x2x1 = 120 ways that exactly the same batch of DVDs can be ordered Therefore the 720 permutations needs tobe divided by 120 to get the answer. Low and behold 720 / 120 = 6 (That we could see just by looking at it!)_ b) Combinations 4 from 8 Maths formula = 8! / 4!4! Permutations = 8x7x6x5 = 1680 Each set of 4 DVDs can be arranged in 4! ways ie 4x3x2x1 = 24 ways 1680 /24 = 70 different combinations (Please note that to explain why say 12!/ 9! = 12 x 11 x 10 when working out permutations - it is for ease of notation. If for instance you had to select 12 from 20 then you would have to write out 20x19x18x17x......x12x11x10x9 every time. This becomes time consuming. Therefore noting that 20! includes all the above numbers multiplied together but also x8x7x6x5x4x3x2x1 if 20! is divided by 8x7x6x5x4x3x2x1 it is the same thing as we require . x8x7x6x5x4x3x2x1 is the same thing as 8!. Therefore 12 from 20 = 20! / 8! and 3 from 12 likewise is written 12! / 9!) In the last question 4 from 8 is 8! /4! (and the second of the 4! relates to the number of permutations each combination can be arranged in)|||You're welcome Report Abuse