Logic Questions with Pigeonhole Principle

Question

"If $n$ objects are distributed into $k$ boxes, then at least one box must contain at least _____ objects." Fill in the blank

How many people must we have in a room to ensure that six of them were born in the same month?

For the first question is it just $k-n$ objects? I dont think it is because it says at least.

Then for the next one is it just $12*5+1$? because it says to ensure so that means we must have it?

Answer 1

The answer to the first question is $E(\frac{n}{k})$ with $E()$ the ceiling function. the second question is now easy.

Answer 2

Your answer to the second question is correct: if you have 60 people in the room, you could have 5 people born each month. You need one additional person to make sure at least one month has at least 6.

Now think about your first question again. The people are the objects ($n$), and the months are the boxes ($k$).