Fundamental Principle of Counting

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Published: 06th November 2012
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In general there are two fundamental principle of counting. These are addition principle and multiplication principle.

The rule of sum: If a first task can be performed in m ways, while a second task can be performed in n ways, and the tasks cannot be performed simultaneously, then performing either task can be accomplished in any one of m+n ways.

The boss assigns 12 employees to two committees.
- Committee A consists of five members.
- Committee B consists of seven members.
- If the boss speak to just one member before making a decision, ?
- If he speak to one member of committee A on the first day, and another member of committee B on the second day, ?

If a procedure can be broken down into first and second stages, and if there are m possible outcomes for the first stage and if, for each of these outcomes, there are n possible outcomes for the second stage, then the total procedure can be carried out, in the designated order, in mn ways.

A license plate consists of two letters followed by four digits.
- If no letter or digit can be repeated?
- If repetitions of letters and digits are allowed?
- If repetitions of letters and digits are allowed, how many of the plates have only vowels (A, E, I, O, U) and even digits?

- n factorial is defined by (a) 0!=1; n!=n(n-1)(n-2)(2)(1)
- Given a collection of n distinct objects, any (linear) arrangement of these objects is called a permutation of the collection
The number of permutations of size r from a collection of n distinct objects is P(n, r)=n!/(n-r)!.

Addition Principle:

Fundamental principle of counting in an event can occur in m different way and another event can occur in n different ways then either of the two events can occur in (m+n) ways provided only one event can occur at a time.

Multiplication principle:

Fundamental principle of counting in an event can occur in m different ways and if corresponding to each way of occurring there are n different ways of the second operation then both the operations can occur simultaneousely in (mn) ways.

Example of Multiplication Principle and Factorial Notation:

For example consider a cinema hall with 4 entrance and 5 exits. Therefore , the number of ways that a person can enter and exit from the cinema hall is (4x5) = 20. Similarly, the number of ways that a person can either enter or out from the cinema hall is (4+5) = 9.

Note: Both these principle can be generalized if in addition principle a third event can occur in p ways then the three events can occur in (m + n + p) ways. Similarly, in multiplication principle the number of ways in which three events can occur is (m n p).

Factorial Notation: Fundamental principle of counting is the continued product of first n natural number is denoted as n and read as factorization mathematically.

N! = 1.2.3.4.5.,,,(n-1).n

= {1.2.3.4.5.(n-1)}.n

=(n-1)! X n.

For example 4! = 4.3.2.1 = 24 and 3! = 1.2.3 = 6

Therefore, it is clear that 4! = 3!*4 = 6*4 = 24

Permutation for Fundamental Principle of Counting

Permutation : Fundamental principle of counting is the number of different arrangement that can be made out of the given number of objects taking some or all at a time is called the permutation.

Permutation with Repetition : Fundamental principle of counting under there is no restriction on the number of times a particular element may occur in r-permutation of n objects. This implies that a repetition is allowed all the r places to be filled up by any of the n objects . Therefore , by multiplication principle the total number of ways given as

N x n x n x n x n x . X n (r factors) = nr.

For example, consider 4 books to be given to 5 student. Hence it is clear that 1st books to be taken by any of the 5 students and similarly 2nd book can be taken by any of the 5 students and so on . Using multiplication principle the total number of possible ways.

= 5 x 5 x 5 x 5 = 625.

Learn more on about Volume of a Rectangular Prism - http://www.mathcaptain.com/geometry/rectangular-prism.html and its Examples. Between, if you have problem on these topics Coterminal Angles - http://www.mathcaptain.com/geometry/coterminal-angles.html, keep checking my articles i will try to help you. Please share your comments.

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