Amy and Frank had a pizza party on Friday night and they had some left

Question

Amy and Frank had a pizza party on Friday night and they had some leftovers. They decide to eat those leftovers for lunch on Saturday. There are 4 pizza slices all with different toppings and 2 bottles of drinks – 1 Coke and 1 Fanta. In how many different ways can Amy and Frank eat the leftovers if they eat one pizza slice and 1 drink each?

A) 10
B) 12
C) 14
D) 24
E) 32

A) 10
B) 12
C) 14
D) 24
E) 32

Kritisood · Answer

Hi  Bunuel   IanStewart   VeritasKarishma   EgmatQuantExpert

In another combinatorics question :  https://gmatclub.com/forum/in-how-many-different-ways-can-a-group-of-8-people-be-divide-85707.html  we divide the combinations by 4! because the number of different orders we can put (the team in) doesn't matter. Similarly, here the answer is 24. But on similar lines, it shouldn't matter whether Amy eats the pizza first or Frank. Hence, why are we not diving by 2! here?

P.s after doing the question in the link, I have been very confused regarding this concept. I would really appreciate some help!!

RTarat · Answer

There are 12 ways in which both of them can choose 2 slices of pizza out of the 4 available slices.
Now we need to factor in the soft drinks. 
This is a permutation question - one choosing Fanta and the other coke is a different case when one chooses the other way round. 
So there are 2 combinations how both of them choose the drinks.
Thus , 12*2=24 ways

Posted from my mobile device

nick1816 · Answer

Kritisood

If you have to divide things in similar groups, then order doesn't matter. But if you have to divide things in different groups, order will matter.

For example -  https://gmatclub.com/forum/in-how-many-different-ways-can-a-group-of-8-people-be-divide-85707.html 
In this question, you gotta divide 8 people in 4 similar groups. Hence, order doesn't matter here, you gotta divide by 4!

Now let's change the question a bit.
 
In how many different ways can a group of 8 people be divided into 3 teams such that 2 teams comprise of 3 people and 1 team of 2 people.

In this question, 2 teams are similar and 1 is different. Hence, you gotta divide by 2!.

Now come back to this problem

Suppose we have 4 toppings- t_1, t_2, t_3 and t_4 and 2 cold drinks- c_1 and c_2

A (t_1,c_1) and F(t_2, c_2) will be counted different from A(t_2,c_2) and F(t_1,c_1). Hence, you don't have to divide by 2! because order matters in this question.

Kritisood · Answer

Hi Nick, thanks for the explanation!! I am still a little confused though. what do you exactly mean when you say  similar groups  (highlighted above). Does it mean number of people in each group is the same?

In the question at hand, if we didn't have cold drinks then would the groups be similar? all toppings?

Also, in the  above example  (highlighted in blue) groups are different hence order SHOULD matter right? hence  we shouldn't divide by 2!  ?

IanStewart · Answer

I'd strongly advise you to study official questions that test these concepts, because those questions will be worded properly. This question is not - what counts as a "way to eat leftovers"? That phrase doesn't make sense. I can guess the intention behind the question, because I've done a lot of these types of questions before, but on reading the problem, I don't even know if we should count drinking the Coke first, then eating the pizza, as a different "way" to eat than eating the pizza first and then drinking the Coke.

They mean to ask in how many ways Amy and Frank can select a pizza slice and drink each. Amy has 4 choices for the pizza, and then Frank has 3 choices left. Amy has 2 choices for a drink and Frank has 1 choice left. Multiplying the number of choices we have, there are (4)(3)(2)(1) = 24 possibilities in total.

Order matters here, because if Amy gets the mushroom pizza and Frank gets the pineapple pizza, that's different from when Amy gets the pineapple and Frank gets the mushroom. So there's no reason to divide by 2!. If instead the question said "Amy will pick two slices of pizza to put in the freezer, and two to put in the fridge", then we'd divide by 2!, because we're just selecting a set of two slices to go in the freezer -- whether we put the pineapple in the freezer first, then the mushroom, or in the other sequence, we still end up with the same two slices in the freezer.

If you revisit the teams question you linked to above, I explained the issue of order there in a similar way. If the teams are in a particular order (1st Team, 2nd Team, etc), or are labelled in some other way (Olympic Team, Davis Cup Team, etc), then order matters, and you do not divide by 4!. If the teams are unlabelled, though, then their order does not matter. That teams question is, incidentally, nothing like any official counting problem I've seen, so I wouldn't worry about it too much, though it might be a useful question to study if you really want to understand the issue of order in counting.

Kritisood · Answer

Thanks  IanStewart  this really helps. I guess my basics just weren't strong enough. I was really struggling after having read the question in the link and was getting easier questions (like this one) also wrong cus of overthinking the problem.

BlackSky · Answer

No of ways: Amy can select any 1 slice and any 1 drink for which Frank can do the likewise. Order doesn’t matter. 
No of ways- 4C1*2C1 * 3C1*1C1=24

Would be great to get a second opinion on this method from any other expert.

Posted from my mobile device

ScottTargetTestPrep · Answer

The number of ways they can eat the leftovers is:

4P2 x 2P2 = 12 x 2 = 24

Answer: D

sayan640 · Answer

Instead of using the formulae use the counting principle. Amy has 4 different choices of pizza , say , P1 , P2 , P3 and P4.
If Amy goes for the P1 pizza , he has 2 choices of drinks i.e coke and fanta.
Similar ly , if Amy goes for P2 pizza, he has 2 choices of drinks i.e coke and fanta.
 So , in total, he is having 2 + 2+2 +2 = 8 choices for eating pizza s and drinks.

Once , Amy selects a pizza and a drink , then it's frank's turn.
Amy has already selected a pizza. So Frank has 3 choice s pizza and 1 choice of drink.
 So in total , frank is having 1+1+1 = 3 choices for eating pizza s and drinks.

So , in total , Amy and Frank eat the leftovers if they eat one pizza slice and 1 drink each , in 8 *3 = 24 ways.

D is the answer.  egmat   KarishmaB   gmatophobia

Amy and Frank had a pizza party on Friday night and they had some left

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