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Premutation and Multiplication Principles. Done by OWF, c October 31, 2013. Multiplication principles. This principle looks at the possible number ( R n ) of outcomes (O n ) that can be derived. Multiplication Principles. - PowerPoint PPT Presentation
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PREMUTATION AND MULTIPLICATION PRINCIPLES
Done by OWF, c October 31, 2013
MULTIPLICATION PRINCIPLES
This principle looks at the possible number (Rn) of outcomes (On) that can be derived
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.
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
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.
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
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?
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
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:
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!
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
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)!
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 nPr to denote the function that evaluates the permutation symbol.