Permutations & Combinations: Introduction

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In Permutations & Combinations, we are only concerned about the number of ways to accomplish a task which may be itself composed of several sub-tasks.

 The entire subject is based on the Fundamental Principle of Counting(FPC).

The basics of Permutations and combinations are counting, it is not concerned in particular ways to accomplish a task.
Independent events: Two events are independent if ocuurence of one cannot affect the occurence of other or the outcome of the event A has no effect on the outcome of event B.
We often get confuse between product rule and sum rule while solving the problems of permutation and combination. Below, i will make it easy for you:

Product rule: If a process can be completed by a few sub sequancial independent processes, number of ways of doing the process will be equal to product of number of ways of doing the sub-processes.
e.g.  We need to go from a station A to station C and we have to find the number of ways to do this task. Firstly, we observe from the figure that there are 3 ways to go to B from A.

 If we follow any one path from A to B it will not affect our decision to choose the path from B to C. So these two events i.e. to choose the path from A to B and from B to C are independent events. 

 Further, for every path from A to B there are 4 choices from B to C. We have 3 choices from A to B and for every choice there are 4 choices from B to C

 Therefore, total number of ways to reach C from A = no. of paths from A to B * no. of paths from B to C
                                                                             = 3 * 4 
                                                                             = 12

 The working of product rule is based on above example. If there are several sub-tasks and suppose a sub-task can be done in n1 ways and second sub-task can be done in n2 ways and for every choice of completing a sub task you find n2 number of ways to do other sub-task then you need to put multiplication sign between the number of ways of these two tasks i.e. {n_1} \times {n_2}


 Product: Remeber for every choice of a path for a sub task you have  equal number of choices of paths for second sub task.

 e.g. Find the number of 4-digit words form by the letters 'PALM' ?

 first we can pick any one of 4 letters i.e. we have 4 choices, after choosing one letter now  3 letters left so 3 choices , then 2 and finally only 1 choice left.

 so, for four letter words , choices for every blanks are:

 4   3   2   1

 now for every letter choose in first place we have 3 choices in 2nd place and for every letter of second place we have 2 choices in 3rd place. So these are independent events and we need to take product to get required number of ways:

4*3*2*1
= 24

Mutually Exclusive Events: Occurence of one event prevents the occurence of others or If event A happens, B cannot and vice-versa.

Sum Rule: The process can be completed by parallel mutually exclusive sub-processes, number of doing the process will be the sum of number of ways of doing the sub process.
In the above figure there are two more different ways directly from A to C. So, if you choose one parallel path you cannot choose another path of A to B to C. So, these are mutually exclusive events so we will apply sum rule.

So, total number of ways to reach C from A = 3*4 + 1 + 1 = 14



PROBLEMS:

Now, in this topic problems are based on 

 *Order Important ( Arrangement )
*Order Unimportant (Selection)
*No Repetition
*Repetition

So, there are basically four types of problems:

 1. Arrangement without Repetition ^n{P_r}

 2. Selection without Repetition  ^n{C_r}

 3. Arrangement with infinite Repetition, {n^r}

 4. Selection with infinite Repetition, ^n{H_r} or ^{n + r - 1}{C_r}


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