PERMUTATION AND COMBINATION (PART–2)

• FUNDAMENTAL RULE OF MULTIPLICATION

Suppose there are three platforms at a railway station namely PLATFORM-I, PLATFORM-II and PLATFORM-III. There are three ways/routes namely A,B, and C through which a person can go from PLATFORM-I to PLATFORM-II. While for going from PLATFORM-II to PLATFORM-III, there are four ways/routes namely 1, 2, 3 and 4 (see the picture below).

Now, suppose you have to find the number of ways/routes through which you can move from PLATFORM-I to PLATFORM-III, via PLATFORM-II.
How Can you do that?
You can take the root A-1
                       or, root A-2
                        or, root B-3 ………….and so on.
Actually there are two tasks you will have to do.In
First task i.e.; to move from PLATFORM-I to PLATFORM-II and that can be done in 3 ways.
Second task i.e.; to move from PLATFORM-II to PLATFORM-III and that can be done in 4 ways.
Now, according to fundamental rule of multiplication if  you have ‘m’ number of ways of doing the first task and corresponding to any one way here you have ‘n’ ways of doing the second task. So the total number of ways in which you can do both the tasks simultaneously is m * n
So total number of ways through which you can move from PLATFORM-I to PLATFORM-III
= 3 * 4
= 12 ways Ans.

• FUNDAMENTAL RULE OF ADDITION

Suppose there are three platforms at a railway station namely PLATFORM-I, PLATFORM-II and PLATFORM-III. There are three ways/routes namely A,B, and C through which a person can go from PLATFORM-I to PLATFORM-II. While for going to PLATFORM-III from PLATFORM-I there are four ways.
 
Now, Suppose you have to go from PLATFORM-I to either PLATFORM-II or to PLATFORM-III directly. In how many ways you can do it?
Here it must be noted that here you have to perform only one task i.e. to move from PLATFORM-I  to any one of the platforms II or III.
Moving from PLATFORM I to PLATFORM II can be done in 3 ways,
While moving from PLATFORM I to PLATFORM III can be done in 4 ways,
So, the total number of ways in which it can be done
= 3 + 4
= 7 Ans.
Example 1: How many numbers of 4 digits can be formed by using the four digits 3, 4, 5 and 6, under the condition that repetition is not allowed?
Solution: Repetition is not allowed means you can not use a digit more than a time in a number
e.g. Correct – 3456, 4536,………………
       Incorrect – 3356, 4556,
Always start from the right most digit. The first digit of the number may be either 3 or 4 or 5 or 6. It means there are 4 ways in which the first digit of the possible numbers can be written.
4 *    *     *    
After writing the first digit, we are left with the 3 digits. It mean there are 3 ways in which we can write the second digit.
4 * 3 *      *   
Now we left with only 2 digits, so we can write the third digit of the possible numbers in two ways.
4 * 3 * 2 * 
And at last, we have only one number, so the last digit of the possible numbers can be written in only one way.
4 * 3 * 2 * 1  = 24 numbers Ans.

◦

PERMUTATION AND COMBINATION (PART–2)