Combinations of pieces of pizza with multiple rules

Question

This is a question I had to code in python but I dont understand exactly how it works. Is there some way to show the combinations?

Phillip the pizza banker wants to buy some pizza.

He wishes to purchase 23 pieces of pizza, and the available flavours are pineapple, pepperoni and cabanossi. The only restrictions on the combination of pizza pieces that can be purchased are that the number of pieces of pineapple pizza must be a multiple of 3, and the number of pieces of cabanossi pizza cannot exceed 12. In how many ways can Phillip the pizza banker purchase 23 pieces of pizza?

score 2 · Answer 1 · answered Jul 21 '17 at 12:25

2

It's probably simplest to split up the possibilities according to how many slices of pineapple there are. If there are $3k$ slices of pineapple then there are $23-3k$ slices remaining. If $23-3k\leq 12$ then he can have from $0$ to $23-3k$ cabanossi (so $23-3k+1$ possibilities), whereas if $23-3k>12$ then he can have from $0$ to $12$ cabanossi ($13$ possibilities).

The possible values for $23-3k$ are just the numbers $1$ less than a multiple of $3$, i.e. $2,5,8,11,14,17,20,23$, and these contribute $3,6,9,12,13,13,13,13$ ways respectively. So the answer is $82$.

answered Jul 21 '17 at 12:25

Especially Lime

40,828

your answer seems right , but where is mistake in my logic in my answer posted below?? – Atul Mishra Jul 21 '17 at 12:29
You can't buy $12$ cabanossi if you buy $21$ pineapple, since that is already too much pizza. – Especially Lime Jul 21 '17 at 12:31
Ahh got it !!@EspeciallyLime thankyou for reviewing – Atul Mishra Jul 21 '17 at 12:33
Ok thanks for the fast response. Im just leaving it open for other commenters if they wish. – mykill456 Jul 21 '17 at 12:34

Atul Mishra · Answer 2 · 2017-07-21T12:38:58.637

1

If he buys $3$ pineapple then he can buy one of the number from the set =$\{0,1,2,3,4,5,6,7,8,9,10,11,12\}$ for cabinossi , hence there are $13$ ways for $3$ pineapple

Similarly $13$ ways are for $\{3, 6,9\}$ pineapple

But for $12 $ pineapple, you cannot buy $12$ cabinossi, so there are $12$ ways

Similarly for $\{15,18,21\}$ pineapples, you have $\{9,6,3\}$ ways

So total ways$ 39+12+9+6+3=69 $ ways

If you count $ 0$ as multiple of $ 3,$ then there are $69+13=82 $ ways

edited Jul 21 '17 at 12:38

answered Jul 21 '17 at 12:20

Atul Mishra

3,136

Wouldn't it be 84 because you couldn't have 13 for cabinossi just 12. So like 12*7 = 84? – mykill456 Jul 21 '17 at 12:22
You can also buy 0 cabinossi , it is not mentioned you have to buy all kind of pizzas @mykill456 – Atul Mishra Jul 21 '17 at 12:24
@mykill improved my post – Atul Mishra Jul 21 '17 at 12:40

dromastyx · Answer 3 · 2017-07-21T12:46:56.390

You want to find all $(x,y,z)$ such that $$3x+y+z=23$$ $$y\leq 12$$ $$x,y,z \in \mathbb{N}_0$$

So the number of combinations is the number of integer points on the above plane.

For $x\in \{0,1,2,3\}$ there are $13$ possible $y$-values such that $z\in \mathbb{N}_0$ and the equation is satisfied.

For $x\in \{4,5,6,7\}$ there are $12, 9, 6 $ and $3$ possible $y$-values such that $z\in\mathbb{N}_0$ and equation is satisfied.

So in total there are $$4\cdot13+12+9+6+3=82$$ possibilities.

Combinations of pieces of pizza with multiple rules

3 Answers3