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>> Basic Principles of Counting Part-1
FUNDAMENTAL PRINCIPAL OF COUNTING
Rule of Product
If one experiment has n possible outcomes and another experiment has m possible outcomes, then there are m × 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 ...... then 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 rule of product.
A college offers 7 courses in the morning and 5 in the evening. Find the possible number of choices with the student who wants to study one course in the morning and one in the evening.
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 × 5 = 35.
A man has five friends. In how many ways can be invite one or more of them to a tea party.
Case I: The man can invite one friend in 5C1 ways
Case II: The man can invite two friend in 5C2 ways
Case III: The man can invite three friend in 5C3 ways
Case IV: The man can invite four friend in 5C4 ways
Case V: The man can invite five friend in 5C5 ways
All these five cases are mutually exclusive i.e. the man can either invite one friend or two friends or ............., but can not simultaneously invite one friend and two friends.
Remember: Whenever there is 'OR', we add.
Therefore Total number of ways in which the man can invite his friends
= 5C1 + 5C2 + 5C3 + 5C4 + 5C5 = 5 + 10 + 10 + 5 + 1 = 31
This problem can also be solved as follows:
Each of his friends may be dealt in two ways - either invited or not invited.
Therefore The total number of ways of inviting all friends is
= 2 × 2 × 2 × 2 × 2 = 32 ways.
But this includes the case, in which all the friends are not invited. Hence, the total number of ways in which the man invites one or more friends is 25 - 1 = 31 ways.
| Note: In the above illustration, we have found the total number of combinations of n dissimilar things taking any number of them at a time, which is 2n - 1.