There are 12 children .Assuming there are 4 children’s bedrooms show that there are at least 3 children sleeping in at least one of them.

Question

There are 12 children in the family

Assuming there are 4 children’s bedrooms in the house, show that there are at least 3 children sleeping in at least one of them.

My question is can I use Pigeonhole Principle ? what do I need to explain? I know the solution but I don't quite get it so please explain.

My solution :(not sure how to do it) like there are 12 children(pigeon) 4 bedroom(holes) then each bedroom must have 4 children but it says that at least 3 children sleeping in at least one of them. Then other room has more than 2 children sleeping.

Answer 1

Use the strong form of Pigeonhole principle: if $m$ objects are put in $n$ containers, then at least one of them contains at least $\lfloor \displaystyle\frac{m-1}{n}\rfloor+1$ objects.

Answer 2

One can apply the pigeonhole principle if you break the problem (assigning 12 children to 4 bedrooms) into a sequence of assignments, for all/some of which the pigeonhole principle directly applies. Consider breaking the problem into a sequence of 3 assignments: The 1st being choose some 5 children and assign to 4 bedrooms. The 2nd being choose some 5 children (not from 1) and assign to 4 bedrooms. The 3rd being choose 2 children(not from 1 or 2) and put them in 4 bedrooms.

from there it's just pigeonhole + some counting argument.