Counting Techniques

Our Main tool for counting is The Basic Counting Principle. For some problems, we will have to use this multiplication principle directly.

Following are some of the often used applications of this counting principle:

Situation

Number of ways

Examples

1. Ordered selection of r objects from a collection of n objects - without replacement.

2. Also called PERMUTATION of n objects taken r at a time.

 nPr= n!/(n-r)! = Product of r integers, starting from n - downward.
  1. Assignment of r seats in a row to a group of n guests.
  2. Assignment of top r ranks to a group of n competitors.

 
1. Ordered selection of r objects from a collection of n objects - with replacement. nr
  1. Number of words of length r with English alphabets = 26r.
  2. Numbers of outcomes, if you throw a die r times = 6r.
1. Unordered selection of r objects from a collection of n objects- all in a group. nCr= n!/(r!*(n-r)!)
  1. Number of committees with r members from a group of n people = nCr.
  2. Number of hands of 13 cards from a deck of 52 cards =52C13.