Premutation and Multiplication Principles

1. Premutation and Multiplication Principles Done by OWF, c October 31, 2013

2. Multiplication principles This principle looks at the possible number (Rn) of outcomes (On)that can be derived

3. Multiplication Principles • Multiplication Principles - If one experiment has n possible outcomes and another experiment has m possible outcomes, then there are m × n possible outcomes when both of these experiments are performed. • Example -1: A college offers 7 courses in the morning and 5 in the evening. Find the possible number of choices with the student if he wants to study one course in the morning and one in the evening.

4. 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 the total number of ways in which he can make the choice of one course in the morning and one in the evening = 7 × 5 = 35. • Example -2: A person wants to go from station A to station C via station B. There are three routes from A to B and four routes from B to C. In how many ways can he travel from A to C? • Solution: A –> B in 3 ways B –> C in 4 ways => A –> C in 3 × 4 = 12 ways

5. The rule of product is applicable only when the number of ways of doing each part is independent of each other i.e. corresponding to any method of doing the first part, the other part can be done by any method. • Example -3: 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 is equivalent to filling 5 places. • Places • 1 2 3 4 5 • Number of Choices: • 5 • 4 • 3 • 2 • 1 • The first place can be filled in 5 ways using anyone of the given digits.

6. 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 of filling all the five places • = 5 × 4 × 3 × 2 × 1 = 120 • => 120 5-digit numbers can be formed. • (ii) Making a 3-digit number is equivalent to filling 3 places. • Places: • 1 2 3

8. Premutation • permutation of a set is an ordered arrangement of the objects in the set. With permutations, ORDER MATTERS. • Suppose 4 pictures are to be arranged from left to right on one wall of an art gallery. • How many arrangements are possible?

9. Using the multiplication principle, there are 4 ways of selecting the first picture. After the first picture is selected, there are 3 ways of selecting the second picture. After the first 2 picture is selected, there are 2 ways of selecting the third picture. And after the first 3 pictures are selected, there is only 1 way to select the fourth. Thus, the number of arrangements possible for the 4 pictures is • 4 3 2 1 4! or 24

10. In general, we refer to a particular arrangement, or ordering, of n objects without repetition as a permutation of the n objects. How many permutations of n objects are there? From the reasoning above, there are n ways in which the first object can be chosen, there are n 1 ways in which the second object can be chosen, and so on. Applying the multiplication principle, we have Theorem 1:

11. Theorem 1 • Theorem 1 Permutations of n Objects • The number of permutations of n objects, denoted by Pn,n, is given by • Pn,n n (n 1) . . . 1 n!

12. Now suppose the director of the art gallery decides to use only 2 of the 4 available pictures on the wall, arranged from left to right. How many arrangements of 2 pictures can be formed from the 4? There are 4 ways the first picture can be selected. After selecting the first picture, there are 3 ways the second picture can be selected. Thus, the number of arrangements of 2 pictures from 4 pictures, denoted by P4,2, is given by P4,2 4 3 12

13. Theorem 2 Permutation of n Objects Taken r at a Time • The number of permutations of n objects taken r at a time is given by • Pn,r n(n 1)(n 2) . . . (n r 1) ___________________________ r factors OR • Pn,r = n! T 0 <= r <=n (n r)!

14. Note that if r n, then the number of permutations of n objects taken n at a time is P n,n = n! = n! = n! Recall, 0! =1. (n -n)! 0! which agrees with Theorem 1, as it should. The permutation symbol Pn,r also can be denoted by or P(n, r). Many calculators use nPrto denote the function that evaluates the permutation symbol.