Permutations & Combinations

COUNTING PRINCIPLES

There are two fundamental counting principles viz. Multiplication principle and Addition principle. There are certain other counting principles also as given below:

⋄ Bijection principle

⋄ Inclusion-exclusion principle

Multiplication Principle:

If one experiment has n possible outcomes and another experiment has m possible outcomes, then there are m x n possible outcomes when both of these experiments are performed.
In other words, if a job has n parts and the job will be completed only when each part is completed and the first part can be completed in a1 ways, the second part can be completed in a2 ways and so on …. the nth part can be completed in an ways, then the total number of ways of doing the job is a1a2a3……….. an. This is known as the Multiplication principle.

Example : A college offers 7 courses in the morning and 5 in the evening. Find the possible number of choices with the student if he wants to study one course in the morning and one in the evening.

Solution: The student has seven choices from the morning courses out of which he can select one course in 7 ways.

For the evening course, he has 5 choices out of which he can select one in 5 ways. Hence the total number of ways in which he can make the choice of one course in the morning and one in the evening = 7 x 5 = 35.

Example : A person wants to go from station A to station C via station B. There are three routes from A to B and four routes from B to C. In how many ways can he travel from A to C?

Solution:

A -> B in 3 ways

B -> C in 4 ways

=> A -> C in 3 x 4 = 12 ways

Remark:

⋄ The rule of product is applicable only when the number of ways of doing each part is independent of each other
i.e. corresponding to any method of doing the first part, the other part can be done by any method

Example : How many (i) 5 − digit (ii) 3 − digit numbers can be formed by using 1, 2, 3, 4, 5 without repetition of digits.

Solution: (i) Making a 5-digit number is equivalent to filling 5 places. Places:

Number of Choices:

The first place can be filled in 5 ways using anyone of the given digits.

The second place can be filled in 4 ways using any of the remaining 4 digits.

Similarly, we can fill the 3rd, 4th and 5th place.

No. of ways of filling all the five places

= 5 x 4 x 3 x 2 x 1 = 120 => 120       5-digit numbers can be formed.

(ii) Making a 3-digit number is equivalent to filling 3 places.

:Places

Number of Choices:

Number of ways of filling all the three places = 5 x 4 x 3 = 60

Hence the total possible 3-digit numbers = 60.

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Addition principle

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