I don't know if this is the correct category, but it's an interesting and difficoult problem (at least for me)

[flocs]

You have 200 dice and on 4 faces of them there is a 2, on the other 2 faces there is a 5. You throw all 200 dice at the same time and you sum all the results. What is the number of possible results of the sum? I don't know if I've transalated well the text because I'm italian and the problem was written in italian, but I've tried my best.

There are 5 possible choices: a)201 b)600 c)301 d)200 e)401

Thank you in advance and sorry for my bad english.

 

[lev888]

You have 200 dice and on 4 faces of them there is a 2, on the other 2 faces there is a 5. You throw all 200 dice at the same time and you sum all the results. What is the number of possible results of the sum? I don't know if I've transalated well the text because I'm italian and the problem was written in italian, but I've tried my best.

There are 5 possible choices: a)201 b)600 c)301 d)200 e)401

Thank you in advance and sorry for my bad english.

Since we are talking about all possible sums and not probabilities, it seems we can ignore the extra faces and simply think of 200 coins with a 2 and a 5 on their faces.

 

[flocs]

Since we are talking about all possible sums and not probabilities, it seems we can ignore the extra faces and simply think of 200 coins with a 2 and a 5 on their faces.

Yes exactly and I've tried to solve the problem just like you have said, by simplifying the dice throw in a coin flip, but I still can't solve the problem, maybe I'm missing something or maybe I'm not smart enough : (

 

[Dr.Peterson]

Yes exactly and I've tried to solve the problem just like you have said, by simplifying the dice throw in a coin flip, but I still can't solve the problem, maybe I'm missing something or maybe I'm not smart enough : (

Let's simplify it one more step: Suppose we replaced the numbers with 0 and 3, by subtracting 2 from each face. That won't change the number of possible sums, right? (Can you explain why?)

So how many different sums can you get by adding 200 numbers, each of which might be a 0 or a 3?

 

[flocs]

Let's simplify it one more step: Suppose we replaced the numbers with 0 and 3, by subtracting 2 from each face. That won't change the number of possible sums, right? (Can you explain why?)

So how many different sums can you get by adding 200 numbers, each of which might be a 0 or a 3?

Ohhh I see the solution now. The numbers thet we sum don't change the number of results that we could have, so we can replace 2 and 5 with different numbers (as long as the two numbers are different to each other) and obtain the same result. So if we start from 0x200
and continue adding the numbers up to 3x200 we obtain 201 different results. Am I right?

 

[Dr.Peterson]

Ohhh I see the solution now. The numbers thet we sum don't change the number of results that we could have, so we can replace 2 and 5 with different numbers (as long as the two numbers are different to each other) and obtain the same result. So if we start from 0x200
and continue adding the numbers up to 3x200 we obtain 201 different results. Am I right?

It would be appropriate to prove that you can do what you say (which is why I started with a specific substitution); but the conclusion is correct. Each different number of 5's leads to a different sum.