Calculation of permutations of 'n' objects when all are taken at a time.

Till now, we have learnt that permutations are calculated by counting the total number of different arrangements that can be made by re-ordering a particular group of objects. Now we will learn to apply two concepts, factorial notation and fundamental principle of counting and derive a formula for calculating the number of permutations of 'n' objects, when all are taken at one time.
Suppose there are four objects of different colours. To arrange them in different orders, we can follow the following thought-process:

• Since all four objects are to be arranged in different orders, we can consider it as a group of four empty boxes, in which we have to fill in the four objects in different orders.
• In the event of filling in the first empty box, we have the choice of four objects.
• After we have placed any one of the four objects in the first empty box, we have 3 objects left to fill the next empty box.
• After we again chose and place one of the three objects in the second empty box, there are two objects that can be placed in the third empty box.
• And after filling the third empty box with any one of the two objects left, a single object is left for filling in the fourth empty box.

We changed our view point about permutations now. Now we see each permutation as a combination of events (of filling up the four different boxes here). Now we apply the fundamental counting principle (click here to learn the fundamental counting principle).

The fundamental counting principle states that if an event can occur in 'm' different ways, and if after the occurrence of this event, another event can occur in 'n' different ways, then the total number of ways in which both the events can occur is m x n.

1. In our example, the first event is of placing an object in the first box. This can occur in four different ways, since we can place any of the four objects in the box.
2. The second event is the placing of an object in the second box. This can occur in three different ways, since we can place any of the three objects in the box.
3. The third event can occur in two different ways since there are two objects to chose from.
4. The fourth event can occur in only 1 way, as only 1 object is left after filling in the first three boxes.

Therefore the total number of different ways in which the four events can occur is given by
4 x 3 x 2 x 1 = 24. (this is the fundamental principle of counting)
Thus, there are 24 different ways in which four objects can be rearranged among themselves. That is, the total number of permutations of 4 object is equal to 24.

By applying the fundamental counting principle in permutations, we obtain that for 4 objects, total number of permutations are given by 4 x 3 x 2 x 1 = 24.Similarly, for 5 objects, the total number of permutations is given by 5 x 4 x 3 x 2 x 1 = 120.

Thus, for any number 'n', the total number of permutations is given by

n x (n - 1) x (n - 2) x ........ x 3 x 2 x 1

By applying factorial notation (click here to study factorial notation) to the above statement, we get

n! (factorial of n) = n x (n - 1) x (n - 2) x ........ x 3 x 2 x 1

Thus, the total number of permutations of 'n' number of objects is equal to the factorial of the number n.

Solved examples:

1. Calculate the total number of permutations when 15 objects are to be arranged in different ways.

Solution:

Total number of objects, n = 15

Total number of permutations, n! = 15! = 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 1307674368000

Therefore, the total number of permutations of 15 objects to be arranged in different ways is 1307674368000.

2. Calculate the total number of ways in which 10 people can be seated in a row.

Solution:

Total number of people, n = 10

Total number of permutations, n! = 10! = 10 x 9 x 8 x 7x 6 x 5 x 4 x 3 x 2 x 1 = 3628800

Therefore, the total number of ways in which 10 people can be seated in a row is 3628800.

3. How many different words can be formed with the first ten letters of the English alphabet given that no letter should be repeated in a word?

Solution:

Total number of letters taken, n = 10

Total number of words formed = permutations of 10 letters = 10! = 3628800