Permutation and Combinations

The simplest sort of permutations question might ask you how many different arrangements are possible for 6 different chairs in a row, or how many different 5 letter arrangements of the letters in the word SMOKE are possible. Both of these simple questions can be answered with the same technique.

Factorial

Factorial is nothing more than a notation represented by the sign!. A factorial is simply the product of a series of integers counting down to 1 from the specified number.

download

6! is read as 6 factorial and it means 6.5.4.3.2.1, which equals 720. The number of possible arrangements of any group with n numbers is simply n!. In this way, the number of possible arrangements of the letters in SMOKE is 5!. Because there are 5 letters in the group, That means 5.4.3.2.1 arrangements, or 120.

Permutations in Small Groups

Permutations get a little difficult when you work with smaller arrangements. For example, what if you were asked how many 2- letter arrangements could be made from the letters in SMOKE?

Here the total number of elements in the letter SMOKE is 5, and the number of elements in the group is 2

You can solve this problem in quite a simple relation by introducing the terms .n = 5 and r = 2

Total Possible Arrangements = $\genfrac{}{}{1px}{}{\mathrm{n!}}{\mathrm{(n\; -\; r)!}}$

Hence in this case, the total arrangement =

Two Tasks Possibilities

If an operation can be performed in ‘m’ different ways and then a second operation can be performed in ‘n’ different ways, then the two operations taken together can be performed in ways. This can be extended to any finite number of operations.

Example:

A hall has 6 gates. In how many ways can a man enter the hall through one gate and come out through a different gate?

Since there are 6 ways of entering into the hall, therefore for coming out the hall through a different gate, number of ways = 5

Hence by the fundamental principle of multiplication, the total number of ways.

Two Independent Tasks Possibilities

If an operation can be performed in ‘m’ different ways and another operation, which is independent of the first operation, can be performed in ‘n’ different ways, then either of the two operations can be performed in ways. This can be extended to any finite number of mutually exclusive events.

Example

There are 30 students in a class in which there are 20 boys and 10 girls. The class teacher selects either a boy or a girl for monitor of the class. In how many ways the class teacher can make this selection? Clearly, there are 20 ways to select a boy and 10 ways to select a girl. Number of ways.

Combinations

Combination is another form of arrangement problem. It differs from permutations in just one way: In combination, the order doesn’t matter. A permutations question might ask you to form different numbers from a set of digits. The order would certainly matter in that case, because 135 is very different from 513. Similarly, a question about seating arrangements would be a permutations question, because the word “arrangements” tells you that order is important. So questions that ask about ”schedules” or “orderings” require you to calculate the number of permutations.

Combinations questions, on the other hand, deal with groupings in which order isn’t important. The group of Kausar - Laila – Aamir isn’t any different from Aamir – Laila – Kausar, far as committees go, in the same way. Combinations questions often deal with the selection of committees, teams, or pairs. Combination and permutation question can be very similar in appearance. Always ask yourself carefully whether a sequence is important in a certain question before you proceed.

Calculating Combinations

Calculating combinations is quite easy. All you have to do is throw out duplicate answers that count as separate permutations, but not as separate combinations. You, if more mathematical mind might use the relation:

Total Possible combinations = $\genfrac{}{}{1px}{}{\mathrm{n!}}{\mathrm{r!(n\; -\; r)!}}$