Fundamental principle of counting
If an event can occur in m different ways, following which another event can occur in n different ways, then the total number of occurrence of the events in the given order is
Permutations
Combinations

^{n}P_{r} = ^{n}C_{r} r!

^{n}_{ }=

^{n} = ^{n}

If ^{n}_{ } = ^{n}
Examples
Question
How many 2 digit even numbers can be formed from the digits 1, 2, 3, 4, 5 if the digits can be repeated?
Solution
There will be as many ways as there are ways of filling 2 vacant places in succession by the five given digits. Here, in this case, we start filling in unit's place, because the options for this place are 2 and 4 only and this can be done in 2 ways; following which the ten's place can be filled by any of the 5 digits in 5 different ways as the digits can be repeated. Therefore, by the multiplication principle, the required number of two digits even numbers is 2 × 5, i.e., 10.
Question
Find the number of permutations of the letters of the word ALLAHABAD.
Solution
Here, there are 9 objects (letters) of which there are 4A's, 2 L's and rest are all different.
Therefore, the required number of arrangements =
Question
In how many ways can 4 red, 3 yellow and 2 green discs be arranged in a row if the discs of the same colour are indistinguishable?
Solution
Total numbers of discs are 4 + 3 + 2 = 9. Out of 9 discs, 4 are of the first kind (red), 3 are of the second kind (yellow) and 2 are of the third kind (green).
Therefore, the number of arrangements
Question
A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how many ways can this be done? How many of these committees would consist of 1 man and 2 women?
Solution
Here, order does not matter. Therefore, we need to count combinations. There will be as many committees as there are combinations of 5 different persons taken 3 at a time. Hence, the required number of ways = ^{5}C_{3} = 10.
Now, 1 man can be selected from 2 men in ^{2}C_{1} ways and 2 women can be selected from 3 women in ^{3}C_{2} ways.
Therefore, the required number of committees = 2C1_{ }* ^{3}C_{2 }= 6.