Understanding Permutation & Combination
IIM Lucknow | MathOratory
First thing that needs to be understood about Permutation & Combination, is that these are primarily methods of counting ways of doing a certain activity (in a structured manner).
For counting, the basics at the end of the day is to understand when to add an when to multiply cases. Let us understand this difference with a very simple example. Let's say a husband and a wife goes to a certain party. The husband has 5 shirts and 6 trousers to select from. The wife has 5 sarees and 6 salwars to select from.
How many ways can they dress up? The husband has to wear both a shirt and a trouser. The wife, however, will choose from wearing either a saree or a salwar. Now, understand this logically. If the man needs to wear both, then they are corresponding events. Let us understand what happens in corresponding events. For each shirt, there are 6 choices for trousers. So, in this case we should multiply 5 and 6. Now, for the lady, we have an option of either a saree or a salwar. So, they are different cases. If they are different cases, we should add 5 and 6.
Thus, if all events have to happen (and types), then they are corresponding events and the cases for the events need to be multiplied.
If either of the events have to happen (or types), then they are different events and cases have to be added
This is the most important first learning of PnC. Please practice this understanding over simplest of problems and lot of problems, again and again. For each basic question, first try to understand whether the events are corresponding or different. Consciously practice, till it comes automatically.
Next thing that needs to be understood is the difference between selection and arrangement. The only difference between selection and arrangement is ORDERING. Ordering is basically the order of placing the items one after the other. In selection, obviously there is no requirement of placement. So selection is without ordering. Arrangement however takes into account the ordering.
If, we apply the similar understanding as above, arrangement is basically the event of first selecting and then ordering them (provided that for each set of selection, we have an equal number of ordering). Then, applying the concept of corresponding events, in case each selection has equal number of ordering, then the number of arrangement = selection * ordering.
This, breeds the formula of nPr = nCr * r!, where nPr is the number of ways of arranging n items over r places, and nCr is the number of ways of selecting r items out of n choices.
I always tell my students, as my teachers used to tell me, that perspective matters. In case of PnC, it matters a lot. Let's compare two problems.
Problem 1. In how many ways can we make 10 people sit over 5 seats
Problem 2. In how many ways can we place 10 distinct marbles in 5 bowls.
The difference between the above two problems is that in the first one, each seat can be occupied by only 1 person. But, in the second one, we can have any bowl holding more than one marble. Another difference is that each seat must be occupied in the 1st problem, but in the 2nd one, a bowl might be left empty.
Solution 1. By experience I can say that this question will be easier to solve if we take the perspective of the seats. Why? If we take the perspective of the people, the first person may or may not be seated. But if you take the perspective of the seats it is easier to solve. The first seat can be occupied by any of the 10 people, the second one can be occupied by any of the remaining nine and so on. So the number of arrangements should be 10 * 9 * 8 * 7 * 6
Solution 2. In this question however if we take the perspective of the bowls, then it will be very complicated. Let us understand why. The first bowl will have 2^10 choices, as every marble may or may not be put in it. But for each choice, then next bowl won't have the same number of choices. As it will depend on how many marbles have gone in the first one. But now if we take the perspective of the marbles, each of them can go to one of the five bowls. And irrespective of where the first marble goes, the next will again have a choice of 5 bowls. So, in this case the answer 5^10.
If you keep this simple, yet logical things in mind, I've no doubt in a few days, you'd also agree with me when I say that PnC is the most logical chapter in maths.