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

Prove by induction that:

ni=

Σ

i=0n(n+1)2Prove by contradiction that √2 is irrational.

How many odd numbers are there between 100 and 999 (inclusive)? How many of these have distinct digits?

You buy 40 different items at a supermarket. You take 13 of them, and three friends with you take 9 each. How many different ways can these be distributed?

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 exaclty? How many of those have at least 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.