Permutations and Combinations
2. Fundamental Principle of Counting
Fundamental Principle of Counting
(a) Multiplication Rule
Suppose there are two independent events A and B, such that event A can be done in \(m\) ways and a second event B can be done independently in \(n\) ways, then both the events A and B can be done in \(m\times n\) ways. Suppose there are 3 ways of reaching from Jaipur to Delhi and 4 ways of reaching from Delhi to Agra, then the number of ways of reaching Jaipur to Agra via Delhi will be 3×4 = 12.
Let us take another example. Suppose we want to find the number of factors of the number 18. We know that 18 can be written as \({2^1} \times {3^2}\). We see that any number of the type \({2^m} \times {3^n}\)will be a factor of 18, where \(m\) can take a value 0 or 1 and \(n\) can take any value from 0, 1, or 2. So the total number of factors will be \(2 \times 3 = 6\). These factors are \(1, 2, 3, 6, 9\) and \(18\).
(b) Addition Rule
If events A and B are mutually exclusive, then both events can be accomplished in m+n ways. Exclusive events are those that cannot happen simultaneously. For example, if a thief can enter a room through 4 doors or 5 windows, then the total number of ways
in which he can enter the room is not 4×5 but 4 + 5, as both cases are mutually exclusive.
Example 01:There are 50 questions on a question paper. In how many ways can a student attempt the paper? They may choose to attempt or not attempt any question from the paper.
Total number of ways of attempting the paper is \(2 \times 2 \times 2 \times 2....... \times {2_{{\rm{(50}}\,\,{\rm{times)}}}} = {2^{50}}\)
Example 02: How many three-digit numbers can be formed using the digits 1, 2, 3, 4,... 9 such that no two consecutive digits are equal?
The third place should be different from the second place, but it can be the same as the first place, so it has 8 choices.
Total number of ways = 9 × 8 × 8 = 576.
Practice Questions