Another combinatorics problem pizzas

Agnishom · 2014-03-02 17:36:40

My problem is below in post #6.

bobbym · 2014-03-02 20:18:24

Hi;

If you must spend the whole $200HKD I am getting 252 as the answer.

Dinner almost done, see you later.

Agnishom · 2014-03-02 23:18:47

What if $200 is the upper bound?

bobbym · 2014-03-02 23:29:58

What do the topping and the pizza cost then? Problem is done in the same way. I used a gf to solve it.

But before that, is the answer correct? Have I interpreted the problem correctly?

Because if you can spend less than $200HKD then the answer is 420.

ElainaVW · 2014-03-03 01:17:37

Hello administrator;

The problem is from another site and is active please cover it up.

Hello Agnishom;

bobbym tells me you are learning M:) Then this is for you.

I computed the answer and used the number of terms in the generating function as a cute test

ans4 = Tuples[{{thk, thn}, {a, b, c, d, e, f}, {a, b, c, d, e, f}, {a,
      b, c, d, e, f}, {a, b, c, d, e, f}}];
ans4 = Union[Sort[#] & /@ ans4];
Length[ans4]
List @@ Expand[(thk + thn) (a + b + c + d + e + f)^4] // Length

252

ans3 = Tuples[{{thk, thn}, {a, b, c, d, e, f}, {a, b, c, d, e, f}, {a,
      b, c, d, e, f}}];
ans3 = Union[Sort[#] & /@ ans3];
Length[ans3]
List @@ Expand[(thk + thn) (a + b + c + d + e + f)^3] // Length

112

ans2 = Tuples[{{thk, thn}, {a, b, c, d, e, f}, {a, b, c, d, e, f}}];
ans2 = Union[Sort[#] & /@ ans2];
Length[ans2]
List @@ Expand[(thk + thn) (a + b + c + d + e + f)^2] // Length

42

ans1 = Tuples[{{thk, thn}, {a, b, c, d, e, f}}];
ans1 = Union[Sort[#] & /@ ans1];
Length[ans1]
List @@ Expand[(thk + thn) (a + b + c + d + e + f)^1] // Length

12

ans0 = Tuples[{{thk, thn}}];
ans0 = Union[Sort[#] & /@ ans0];
Length[ans0]
List @@ Expand[(thk + thn) (a + b + c + d + e + f)^0] // Length

2

252 + 112 + 42 + 12 + 2 = 420

Last edited by ElainaVW (2014-03-03 01:18:41)

bobbym · 2014-03-03 01:28:25

Agnishom · 2014-03-03 01:31:58

What is going on over there? Why are there so many answers?

Brilliant does not have any restriction on sharing their problems on other sites anymore

bobbym · 2014-03-03 01:35:45

I will replace it back if you want but it still would be better to post it in the form of a picture rather than text. Text can be googled for keywords, pictures can not. Many people can solve a tough question now just by looking at our answers. The more people that solve a question, the lower the rating of the question. Also means lower rating for those who solved it like me.

What do you say to this?

Agnishom · 2014-03-03 01:38:10

So, is it morally wrong to post the question here?

bobbym · 2014-03-03 01:43:51

No, if they have removed their restriction then it is okay. I was just recommending posting them as a picture as I have done. I do run into some people who ask those questions on other forums. They are looking to increase their score by cheating. Right now, that does not matter but I would say that soon B will bring back the old system and then points will count again.

There is not many answers just 420 if you can spend 200 or less and 252 if you must spend 200. The question is a little unclear and took me two tries because of that.

Agnishom · 2014-03-03 01:46:56

bobbym wrote:

I do run into some people who ask those questions on other forums. They are looking to increase their score by cheating.

You mean me?

I understand the way it is done with nCr's. But that looks like ugly casework. Is there a way to use generating functions?

Agnishom · 2014-03-03 01:53:03

https://brilliant.org/faq/#solving-problems

See the last question in the Solving Problems section. I did not post this to get an answer

bobbym · 2014-03-03 01:54:02

You mean me?

Of course, I mean you. You varmint! If I were a reppie I would swim to India find you and eat you.

I believe we both did use gf's to solve the problem. I used them to compute the answer she used them as a check. She is perhaps a bit more M inclined then I am. Some of her training was by my brother and the Colonel. They like computers and hate math.

Agnishom · 2014-03-03 02:01:37

Where is the gf? It looks like it is a brute force solution or something.

Please look at Post 12

Who is the colonel?

bobbym · 2014-03-03 02:05:08

The gf is this

The number of terms in that is 252. That is the answer for 200. Please use M to verify that.

Agnishom · 2014-03-03 02:19:09

I will use Maxima instead.

(%i1) nterms(((thk+thn)*(a+b+c+d+e+f)^4))+nterms(((thk+thn)*(a+b+c+d+e+f)^3))+nterms(((thk+thn)*(a+b+c+d+e+f)^2))+nterms(((thk+thn)*(a+b+c+d+e+f)^1))+nterms(((thk+thn)*(a+b+c+d+e+f)^0));
(%o1)                                 420

When are we supposed to be concerned about the number of terms in a gf?

bobbym · 2014-03-03 02:26:25

I would rather you do it in M because an important principle will be learned soon. If you expand her gf you will see that some terms have numbers in front of them for instance.

(4 a^3 b thk) that means in her somewhat cryptic notation thick crust, 3 toppings of a and one of b. We see the number 4 in front.
That 4 means there are 4 ways to arrange a a a b, count them using M. But you notice that aaab and aaba have different order but they are the same pizza because order does not count.
Therefore, every number is discarded leaving 1 of each. That is why she counted terms.

Agnishom · 2014-03-03 02:28:43

I would rather you do it in M ...

Please fix it. The sentence is either not grammatical or not parsable by me

bobbym · 2014-03-03 02:35:40

Instead of using Maxima, doing her work in Mathematica will be more profitable in the long run.

It is also fairly easy to see the NCR solution buried inside here method.

Math Is Fun Forum

#1 2014-03-02 17:36:40

Another combinatorics problem pizzas

#2 2014-03-02 20:18:24

Re: Another combinatorics problem pizzas

#3 2014-03-02 23:18:47

Re: Another combinatorics problem pizzas

#4 2014-03-02 23:29:58

Re: Another combinatorics problem pizzas

#5 2014-03-03 01:17:37

Re: Another combinatorics problem pizzas

#6 2014-03-03 01:28:25

Re: Another combinatorics problem pizzas

#7 2014-03-03 01:31:58

Re: Another combinatorics problem pizzas

#8 2014-03-03 01:35:45

Re: Another combinatorics problem pizzas

#9 2014-03-03 01:38:10

Re: Another combinatorics problem pizzas

#10 2014-03-03 01:43:51

Re: Another combinatorics problem pizzas

#11 2014-03-03 01:46:56

Re: Another combinatorics problem pizzas

#12 2014-03-03 01:53:03

Re: Another combinatorics problem pizzas

#13 2014-03-03 01:54:02

Re: Another combinatorics problem pizzas

#14 2014-03-03 02:01:37

Re: Another combinatorics problem pizzas

#15 2014-03-03 02:05:08

Re: Another combinatorics problem pizzas

#16 2014-03-03 02:19:09

Re: Another combinatorics problem pizzas

#17 2014-03-03 02:26:25

Re: Another combinatorics problem pizzas

#18 2014-03-03 02:28:43

Re: Another combinatorics problem pizzas

#19 2014-03-03 02:35:40

Re: Another combinatorics problem pizzas

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