How many people need to turn up to be sure of making at least one team?

Question

So pigeonhole principle states when there are m objects to be divided into n sets then at least one contains r+1 objects. i.e m > nr In this question the m objects should be 12 months in 5 people so 12>5*3?

How do you arrive at 49?

Answer 1

The value we are trying to find in this problem is $m$, the number of people required to have at least one team of five people.

We want at least one team with $5$ players, so $r + 1 = 5 \implies r = 4$.

The number of teams is the number of months, so $n = 12$.

Hence, according to your formula $m > 12 \cdot 4 = 48$. The smallest such value is $m = 49$.

Answer 2

There are 12 possible months. Let's imagine "the worst" situation for finding team quickly: each new person has the most "unpopular" month of birth - so, ppl are distributed "uniformly" between month. How many ppl you need to have 4 of each type? 4*12=48. And then any new person (+1) will have 1 of 12 possible months of birth and 4+1=5 (team is made). If ppl won't distribute uniformly, then you will need 48 or less (because if one group has 3, then another has 5). 4*12+1=49 is maximum, then.