Combinatorics Problem: Forming Coalitions with at least 5 Mandates

In summary, there are 7 parties competing in the elections with one party having 3 mandates and the rest having one each. For part (a), the number of possible coalitions with at least 5 mandates can be calculated using binomial coefficients. For part (b), the number of possible coalitions formed by calling each party in a random order and declaring the coalition complete when it has at least 5 mandates can be described as all the possible combinations of parties without considering the order. This can also be calculated using binomial coefficients.
  • #1
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Homework Statement



There are 7 parties competing in the elections. One party has obtained 3 mandates, and the rest 6 have received one mandate each.
a) How many different coalitions with at least 5 mandates can be formed?
b) Assume that the parties are being called to join each other in a random order, and a coalition is declared ”complete“ as soon as it has at least 5 mandates. How many different coalitions can be formed this way?

Homework Equations





The Attempt at a Solution



For part a) I assume you could calculate the number of coalitions that can be formed with the party with 3 votes and without it. Which I can do, I think. But I don't think that's the most efficient way and would take a while to work out (or maybe I'm just doing it wrong). And that doesn't really help with b.

I'm not asking for the answer at all, just how I would go about tackling this sort of problem.
 
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  • #2
For part (a), you have the right idea. It's also a really fast calculation as long as you know how to answer the question: I have n balls with the numbers 1,...,n on them in a bag. How many different ways can I remove k of them? The answer uses what's called binomial coefficients

For part (b), can you identify what the difference between it and part (a) is?
 
  • #3
Yeah, I got part a, I was just overcomplicating it with something that didn't need to be done. I believe part b to mean you 'call' each party to join the coalition in turn, and if it is greater than or equal to 5, no more parties join.

So, if the parties are A, B, C, D, E, F and G, where A is the one with 3 mandates. You could have
ABC or BCDEA, but not ABCD or BCDEFG
 
  • #4
For part (b), the question simply states

How many different coalitions can be formed this way?

Which doesn't matter what the order is. So ABCD is a possible coalition, as long as the calling order is B,C,D, then A. So what is a simple way to describe all the possible coalitions that can be formed like this
 

1. How do you determine the number of possible coalitions with at least 5 mandates?

The number of possible coalitions with at least 5 mandates can be determined using the combination formula, which is nCr = n! / (r!(n-r)!), where n is the total number of parties and r is the number of mandates. In this problem, n would be the total number of parties and r would be 5.

2. Can a party form multiple coalitions with different mandates?

Yes, a party can form multiple coalitions with different mandates. This is because the formation of coalitions is not exclusive and a party can join or leave different coalitions depending on their interests.

3. How does the number of parties affect the number of possible coalitions?

The number of parties directly affects the number of possible coalitions. The more parties there are, the more potential combinations of coalitions there can be. This means that as the number of parties increases, the number of possible coalitions also increases.

4. Is there a limit to the number of mandates a party can hold in a coalition?

No, there is no limit to the number of mandates a party can hold in a coalition. A party can hold as many mandates as they are able to negotiate for in a coalition. However, in this problem, we are looking for coalitions with at least 5 mandates, so the number of mandates held by a party in a coalition must be 5 or more.

5. How can the concept of "combinatorics" be applied to real-life situations?

The concept of combinatorics, which involves counting and organizing combinations, can be applied to various real-life situations. For example, it can be used to determine the number of possible outcomes in a game or the number of different combinations of ingredients in a recipe. In this problem, combinatorics is used to determine the number of possible coalitions with at least 5 mandates in a political context.

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