Wikipedia:Reference desk/Archives/Mathematics/2021 October 4

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October 4

Election permutation some kind of combinatorics

If each district of the House of Commons had a ballot of the same x parties (I know this is false) how many results are mathematically possible?

Where exactly 1 Lab and the rest Plaid is one result (doesn't matter where it is); exactly 351 Con, 45 Lib, 58 SNP, 3 Green, 3 Plaid, 1 DUP, 1 UUP, 1 SDLP, 1 Fein, 1 Alliance, 1 UKIP and the rest Lab is another result, 324 Lib 326 Lab is another, vice versa is another and so on. Sagittarian Milky Way (talk) 23:58, 4 October 2021 (UTC)

It isn't clear what you're asking. Does each ballot contain a vote for exactly 1 party? 2601:648:8202:350:0:0:0:1598 (talk) 01:28, 5 October 2021 (UTC)

How many ways can the number of seats won list end up? It's got to be less than 651**x, as all the parties have to add up to 650. Sagittarian Milky Way (talk) 02:35, 5 October 2021 (UTC)

Are we supposed to be familiar with the number of districts and the election procedures? You might start by revealing of which country this is the House of Commons. I suggest you rephrase this question in a more abstract setting.  --Lambiam 03:37, 5 October 2021 (UTC)

I didn't know Canada has one too. Sagittarian Milky Way (talk) 03:55, 6 October 2021 (UTC)

Is this an equivalent problem? There is an unlimited supply of coloured balls, but they come in exactly $c$ different colours. A bag can hold precisely $n$ balls. How may different ways are there to fill a bag with these coloured balls? The answer is ${\tbinom {n+c-1}{c-1}}.$  --Lambiam 04:16, 5 October 2021 (UTC)

Maybe? One equivalent problem is "I took a math test with x questions and answered each with a random integer between 0 and y inclusive. How many possible tests are there where the sum of my answers is y?" Sagittarian Milky Way (talk) 03:55, 6 October 2021 (UTC)

There is a bijection between the set of finished tests and the set of filled bags, using $x=c$ and $y=n.$ Think of each question as having been assigned a unique colour. Then, when answering a question, you add as many balls of its colour to the bag as your answer. If the proverbial dog now eats the paper with your finished test, your answer to each question can be reconstructed from the contents of the bag.  --Lambiam 05:10, 6 October 2021 (UTC)

So it doesn't matter in what order you assemble the bag (which would've made the result bigger than an equivalent problem) Sagittarian Milky Way (talk) 13:42, 6 October 2021 (UTC)

I used the term bag as a synonym of multiset.  --Lambiam 14:44, 6 October 2021 (UTC)

You are asking for the number of compositions of 651 (or however many seats there are) into x parts. The stars and bars theorem is the name of the theorem that counts them. --JBL (talk) 11:42, 7 October 2021 (UTC)