Counting Techniques

 1. FUNDAMENTAL PRINCIPLE OF COUNTING

"If an operation can be performed in  ways and if for each of these a second operation can be performed in  ways, then the two operations can be performed together in 

 ways."

Examples:

1. Three items are selected randomly from a manufacturing process. Each item is inspected and classified as defective (D) or non-defective (N). Determine the number of ways this selection
can be done, using:
a. fundamental principle of counting
b. tree diagram

2. A student has to enroll in Math, Physics and Chem. If there are 2 sections in Math 31 (A and
G), 2 sections in Chem 15 (I and E) and 1 section in Phys 11 (J), how many possible schedules can
he make, assuming there is no conflict of schedules?

3. How many 3-digit numbers can be formed from the digits 2, 7, 8, and 9 if
a. no digit is repeated
b. repetition of digits is allowed
c. repetition of digits is allowed and the number is even.

4. In how many ways can you arrange the letters of the word FORGET such that
a. it ends in a vowel
b. it ends in a consonant and starts with a vowel


2. PERMUTATION

A set of objects can be arranged in different ways depending on the number of objects in the set and the number of objects in a particular arrangement. Each ordered arrangement of all or part of a set of objects is called a permutation.

Property 1. 

The number of permutations of n distinct objects is n!

Example:
a. How many distinct permutations can be made from the letters of the word MATH?
b. How many of these permutations start with the letter M?

Property 2. 

The number of permutations of  distinct objects taken  at a time is

Example:
If the three prizes, the first, second and the third prize will be awarded from among 10 equally-qualified students, in how many ways can this be done?

Property 3. 

The number of distinct permutations of n things of which  are of one kind,  of a second kind (or alike), …,  of the  kind, is given by

where 

Example:
How many different ways can 3 red, 4 yellow and 2 blue bulbs be arranged in a string of Christmas tree lights with 9 sockets?


Property 4. 
The number of permutations of n distinct objects arranged in a circle is (n-1)!

Example:
In how many ways can 6 different varieties of gumamela be planted in a circle?


3. COMBINATION

In some cases we are interested in the number of ways of selecting r objects from n distinct objects without regard to order. These selections are called combinations.

Example:
From 4 Mathematicians and 3 Statisticians, find the number of committees of size 3 that can be formed with 2 Mathematicians and 1 Statistician.