A girl has to make pizza with different toppings. There are 8 different toppings. In how many ways can she make pizzas with 2 different toppings? A) 16 b) 56 c) 112 d) 28

4 Answers

Mark Mottian Profile
Mark Mottian answered
Answer: 8 * 7 = 56 different ways!
The correct answer is B.
Linda Jane Profile
Linda Jane answered
The answer is 8 * 7 because the formula is n!/(n-c)! Where n is the total number of things and C is the number chosen in a grouping. In this problem n=8 and c=2 so we get  8!/(8-2)! = (8*7*6*5*4*3*2*1)/(6*5*4*3*2*1) = 8*7=56
thanked the writer.
Oddman
Oddman commented
You have correctly described the formula for the number of permutations. However, we assume the sequence in which the toppings are applied is irrelevant. In that case, we need the formula for combinations, which has another factor of (c!) in the denominator. 56/2 = 28.
Jacob Morin Profile
Jacob Morin answered
Don't cheat on your homework its the wrong way to go
thanked the writer.
Mark Mottian
Mark Mottian commented
The individual needs help becuase he does not how to work out such a problem! If he knew the answer and was just being lazy, their would be no need to ask the question!
len McRue Profile
len McRue answered
To see why this is so you may want to get a bit practical with it! Do a 'cross table' with all the 8 pizza toppings arranged as below:

            1        2        3      4      5      6      7      8
1        (1,1)  (1,2)  (1,3) ......

2        (2,1)

3        (3,1) .
.
From the complete table, you can count the number of distinct pairings of two different toppings as read from the entries. Clearly, this is not an efficient method for bigger numbers and a formula can be learnt and be applied as appropriate. I note that you have been assisted on this front.
thanked the writer.
len McRue
len McRue commented
The formula you need is the 'binomial coefficient' given by n!/(n-k)!k! Where n=8, k=2. The correct answer is 28 not 56 because for example (3,1) and (1,3) denote the same thing, ie a pizza with toppings 1 and 3.
Oddman
Oddman commented
As you have noted, pair (n, m) is regarded as being the same as pair (m, n). Thus your table only needs to be filled in on the main diagonal and above (or below).

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