A few questions in discrete math taken from (or suggested
by) the textbook
Discrete Mathematics by Stephen Barnett, Addison Wesley, 1998.

Prove that √2 is irrational.

You have five dimes, three quarters, two silver dollars and a two-dollar bill. How many ways are there to distribute these amoung your three neices, Sally, Alice and Karen. Some might get nothing. one thing. (Coins of the same type are same; if it's three dimes, it doesn't matter which three.)

How many of these make sure each neice gets at least one coin (or the bill)? (I don't know).

A standard pack of 52 playing cards has four suits with 13 cards each. If one card is selected from each suit, how many 4-card hands can be created? How many of those have two aces?

In a group of 13 or more people, there will always be (at least) two who have the same birth month. Why?

Suppose there is a group of 12 people. Some are friends with each other. Show that there will be two with the same number of friends.