Probability

# Pigeonhole Principle: Level 1 Challenges

It was around 4 in the morning, and I'm all dressed up, ready for school, when the electricity was cut off. Too bad, I haven't put on my socks yet.

I have 2343 pairs of gray socks, 3212 pairs of pink socks and 6525 pairs of blue socks. Everything is mixed in my drawer (I'm a bit of irresponsible, sorry about that.). As there was no light, I was not able to identify the color of the socks. How many of the socks did I need to take to match one pair?

There are 11 objects that have to have to placed in $n$ slots. Find the number of maximum possible slots $n$ such that there are exists at least a single slot in which 3 objects are placed?

Is it true that in any eight composite positive integers not exceeding 360, that at least two are not relatively prime?

A box contains 100 balls of the following colours: 28 red,17 blue, 21 green, 10 white, 12 yellow and 12 black. What is the smallest number $n$ such that any $n$ balls drawn from the box will contain at least 15 balls of the same colour?

In a box there are red and blue balls. If you select a handful of them with eyes closed, you have to grab at least $5$ of them to make sure at least one of them is red and you have to grab at least $10$ of them to make sure both colors appear among the balls selected. How many balls are there in the box?

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