Combinatorics Problem: Forming Coalitions with at least 5 Mandates

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The discussion revolves around calculating the number of coalitions that can be formed among seven political parties, where one party has 3 mandates and the others have 1 each. For part (a), participants suggest using binomial coefficients to efficiently determine the number of coalitions with at least 5 mandates, considering combinations with and without the party holding 3 mandates. In part (b), the focus shifts to the order in which parties join, emphasizing that the coalition is complete as soon as it reaches 5 mandates, regardless of the joining sequence. The conversation highlights the distinction between the two parts of the problem and the importance of understanding coalition formation rules. Overall, the thread seeks strategies for tackling combinatorial problems in political contexts.
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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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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?
 
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
 
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
 

Thread 'Family of lines that are at a distance of 5 from the origin'

I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
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