Picking hats

N people come to a room to attend a conference. They leave their hats outside the room. When the conference ended, the people were in a hurry, so they picked one hat each at random, one after another. Given N is considerably large, find the probability that no one gets his/her own hat.

Courtesy : Prof. Sharad Sane, IIT Bombay.

The "ballot" problem

There are two candidates contesting for a post, A and B. Elections have ended and A got total a votes and B got total b votes, with a > b. Every person voted must vote for either A or B. They shuffled the ballot box and started counting. A is declared winner only if he leads at every point of counting. What is the probability that A will win the elections?

Courtesy : Prof. S.B. Pillai

Connect the pyramids

We have two pyramids, a square-base right pyramid and a triangle-base right pyramid. All the edges of both the pyramids are of equal length. We joined both the pyramids with one of the triangle-faces of one pyramid coinciding with one of the triangle-faces of another. What are the total number of faces of the new solid figure obtained?

Source : Winkler - Mathematical puzzles (Please go through this book. It is really good)

The "Mailbox" problem

We have n mailboxes, which each have a key and all keys are distinct. There is a master key which works for every mailbox.

Now, each key is put in one mailbox, and all mailboxes are locked using the master key.

So, what we have here is n locked mailboxes, which each has a key to one of those n mailboxes.

You want to open all the boxes. To start with, you have to break one box, take key from that, and open its corresponding box. When you find key to first box, you have to break another box and so on.

So, given you can break k boxes, what is the probability that you will be able to open all the mailboxes?

Courtesy: Prof. Sharad S. Sane, IIT Bombay.

Sequence of non-powers of primes

Given a natural number n, prove that we can always find out n consecutive natural numbers such that none of them is a perfect power of any prime.

Courtesy: Vinod reddy

Factorial-base representation

Show that every non-negative integer n can be uniquely represented as

n = a1*(1!) + a2*(2!) + a3*(3!) ...........

where all ai are integers and 0 <= ai <= i for all i.

Courtesy: Prof. Sharad S. Sane, IIT Bombay.