... From Page :: 3 |
Counting Numbers | |
The numbers that we use to count things or objects. Natural Numbers are called counting numbers.
|
Fundamental Counting Principle of Multiplication | |
If a total event can be sub-divided into two or more independent sub-events, then the number of ways in which the total event can be accomplished is given by the product of the number of ways in which each sub-event can be accomplished.
No. of ways in which the total event can be accomplished
⇒ nE = nE1 × nE2 × nE3 × .... |
Illustration » 1 | |
Consider the journey from "New Delhi" to "New York" via "London".
There are four routes from "New Delhi" to "London" and five routes from "London" to "New York". In how many ways can a person travel from "New Delhi" to "New York" via "London"? To find this,
Let us divide the total event of traveling from "New Delhi" to "New York" into two independent sub-events.
[Because the route taken in one part journey is not dependent on the route taken on the other i.e. what route is taken is not influenced by what route has been taken in the other part journey, we can say that the sub-events are independent.]
There are
Four routes from "New Delhi" to "London" and five routes from "London" to "New York"
⇒ The number of ways in which the journey
Therefore,
The number of ways in which the total task of travelling from "New Delhi" to "New York" can be accomplished
= (No. of ways in which the task of traveling from "New Delhi" to "London" (1st sub-event) can be accomplished) × (No. of ways in which the the task of traveling from "London" to "New York" (2nd sub-event) can be accomplished)
Rationale
Let "A", "B", "C" and "D" represent the four routes from "New Delhi" to "London".
Let "1", "2", "3", "4" and "5" be the numbers representing the routes from "London" to "New York". The possibilities can be summarised as
Each choice of the first can be combined with every choice of the second. |
Illustration » 2 | |
Consider the experiment of drawing 9 balls from a bag.
There are 6 blue, 4 red and 7 white balls in the bag. In how many ways can 3 blue, 2 red and 4 white balls be drawn
the number of ways in which the 3 blue, 2 red and 4 white balls are drawn, we divide the total event of drawing the 9 balls into three sub-events To find this
Let us divide the total event of drawing the 9 balls into three independent sub-events.
Therefore,
The number of ways in which the 9 balls can be drawn such that 3 blue, 2 red and 4 white balls are drawn
= (No. of ways in which the 3 blue balls (1st sub-event) can be drawn from the total 6)
× (No. of ways in which the 2 red balls (2nd sub-event) can be drawn from the total 4) × (No. of ways in which the 4 white balls (3rd sub-event) can be drawn from the total 7)
Working Table
The above idea can be represented in a working table as
|
Author Credit : The Edifier | ... Continued Page :: 5 |