Combinatorics Problem: Forming Coalitions with at least 5 Mandates

[abcdef123]

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.

 

[Office_Shredder]

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?

 

[abcdef123]

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

 

[Office_Shredder]

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