Combinatorics and DS

[QuinganZ341]

Hi,

By doing some combinatorics DS problems in Number Properties QB and OG guide, it seems that once I know z teams of x can be formed, then I know that there are y total individuals. For example, if I know there are 126 teams of 5, then I know there were 9 individuals total to start. If I know there are 56 teams of 3, I know there are 8 individuals to start.

Am I right in concluding that the bolded statement is always true?

[tim]

That is correct. I trust you weren't looking for a proof? I'm glad you brought the question here rather than just assuming it to be true. Please keep in mind though that while your rule is true as written, even minor alterations to the rule could make it untrue. Feel free to bring any further questions back here though, and we'll be glad to help!

[RonPurewal]

well, actually this is super easy to prove, by 'reductio ad absurdum' (= try pretending that it's NOT true, and show that the situation is ridiculous / self-contradictory / impossible).

IF there could be more than one 'y'...
(let's call them 'y1' and y2'—just pretend those are subscripts—and let's say y1 is the smaller one)

...then...

... from a pool of 'y1' people, you could make 'z' different teams,
ond
... from a pool of 'y2' people (more people), you could still only make 'z' teams (the same number of teams).

this is impossible, of course—if you add people to the pool, the number of teams of a given size will ALWAYS be greater (since you'll still have all the teams you originally had, and now you can make teams with the extra people too).

since that is impossible, your theorem is a theorem.

[QuinganZ341]

Thank you so much!

[RonPurewal]

sure.