Permutations and Combination: Basic Principles of Counting Rule of Sum If one experiment has n possible outcomes and another has m possible outcomes, then there are (m + n) possible outcomes when exactly one of these experiments is performed. In other words if a job can be done by n methods and by using the first method can be done in a1 ways or by second method in a2 ways and so on ... by the nth method in an ways, then the number of ways to get the job done is (a1 + a2 + ...... + an). Illustration: How many straight lines can be formed from six points, no three of which are collinear? Solution: To form a straight line, we need to select two points out of six points. This can be done in 6C2 ways = 6.5/2.1 = 15 ways. Illustration: In how many ways can a committee of five be formed from amongst four boys and six girls so as to include exactly two girls? Solution: We have to select two girls from six girls and three boys from four boys. Number of ways of selecting girls = 6C2 = 6.5/2.1 = 15 Number of ways of selecting boys = 4C3 = 4 Number of ways of forming the committee = 15 × 4 = 60. Note: Here we have used multiplication rule. Illustration: A college offers 7 courses in the morning and 5 in the evening. Find the number of ways a student can select exactly one course, either in the morning or in the evening. Solution: The student has seven choices from the morning courses out of which he can select one course in 7 ways. For the evening course, he has 5 choices out of which he can select one in 5 ways. Hence he has total number of 7 + 5 = 12 choices. Illustration: How many (i) 5 - digit, (ii) 3 - digit numbers can be formed by using 1, 2, 3, 4, 5 without repetition of digits. Solution: (i) Making a 5-digit number amounts to filling 5 places. Places: table2 Number of Choices: 5 4 3 2 1 The first place can be filled in 5 ways using any of the given digits. The second place can be filled in 4 ways using any of the remaining 4 digits. Similarly, we can fill the 3rd, 4th and 5th place. No. of ways to fill all the five places. = 5 × 4 × 3 × 2 × 1 = 120 => 120 5-digit numbers can be formed. (ii) Making a 3 - digit number amounts to filling 3 places. Places: table3 Number of choices: 5 4 3 Number of ways to fill all the three places = 5 × 4 × 3 = 60. Hence the total possible 3 - digit numbers = 60.
Posted on: Sat, 28 Sep 2013 06:50:30 +0000