para may kopya din ako

1. Permutation &
Combination

2. Introduction
By considering the ratio of the number of desired subsets to
the number of all possible subsets for many games of chance in
the 17th century, the French mathematicians Blaise
Pascal and Pierre de Fermat gave impetus to the development
of combinatorics and probability theory.

3. Important Terms:
 Factorial
The continued product of first n natural numbers is called the “n
factorial”. It is denoted by an exclamatory symbol like n! or |n.
Examples: n! = n × (n – 1) × … × 4 × 3 × 2 × 1
5! = 5 × 4 × 3 × 2 × 1 = 120
Factorial is defined for whole numbers only.
Factorials of proper fractions or negative integers are not defined.
Factorial of zero is defined to be 1. That is, 0! = 1. This is consistent with
the combinatorial interpretation of there being exactly one way to
arrange zero objects, that is, there is a single permutation of zero
elements, namely the empty set φ.

4.  Fundamental Principle of Counting:
(also known as the multiplication rule for counting) If a task can
be performed in n1 ways, and for each of these a second task
can be performed in n2 ways, and for each of the latter a third
task can be performed in n3 ways, ..., and for each of the latter
a kth task can be performed in nk ways, then the entire
sequence of k tasks can be performed in n1 • n2 • n3 • ... •
nk ways.

5. Permutation
 The origin of permutation is the Latin word mutare which
related to change. The per prefix is an emphsis of complete
or extreme change.
 A set of objects in which position (or order) is important.
 A permutation which only exchanges two elements is often
called a transposition. It is known that every permutation can
be found from any other by a string of transpositions. The
notation to switch the first and third element in a set is (1,3).

6.  A permutation is the choice of r things from a set of n things without
replacement and where the order matters.

7. Two types of Permutation:
 Repetition is Allowed: such as the lock above. It could be
"333".
 No Repetition: for example the first three people in a running
race. You can't be first and second.

8. 1. Permutations with Repetition
These are the easiest to calculate.
When we have n things to choose from ... we have n choices each time!
When choosing r of them, the permutations are:
n × n × ... (r times)
So, the formula is simply:
nr,
where n is the number of things to choose from, and we choose r of them
(Repetition allowed, order matters)

9. 2. Permutations without Repetition
where n is the number of things to choose from, and we choose r of them
(No repetition, order matters)

10. Permutations in a circle:
Sometimes, items are arranged in a circle. In this case, we no longer have a
left end and a right end. The first placed item is merely a point of reference
instead of having n choices.
Thus with n distinguishable objects we have (n-1)! arrangements instead of n!.

11. PERMUTATION EXAMPLES:
1. How many ways can we award a 1st, 2nd and 3rd place prize among eight
contestants? (Gold / Silver / Bronze)?
Let’s say:
Gold medal: 8 choices: A B C D E F G H
Silver medal: 7 choices: B C D E F G H.
Bronze medal: 6 choices: C D E F G H.
Using this formula:
So the total number of options is 336

12. 2. Permutation without Repetitions: What is the total number of
possible 4-letter arrangements of the letters m, a, t, h if each letter is
used only once in each arrangement?
Using the formula: Using calculator:
3. How many 3-letter word with or without meaning, can be formed out of the
letters of the word, 'LOGARITHMS', if repetition of letters is not allowed?
The word 'LOGARITHMS' has 10 different letters.
Using calculator: 10P3 or P(10,3) = 720 three-letter word can be formed
4. Permutation in a Circle: How many six colored squares can be formed in a
circle?
C = (n-1)! = (6-1)! = 120

13. Combination
 The word combine, from which combination derives, is built
of the Latin prefix com, for together, and bini a special form
of the Latin root for two, bi. Bini would be similar to our word
pair, and the idea of combining was to put together two at a
time.
 A set of objects in which position (or order) is NOT important.

14.  A combination is the choice of r things from a set of n things without
replacement and where order does not matter.

15. Two types of Combinations:
Remember the order does not matter now.
 Repetition is Allowed: such as coins in your pocket
(5,5,5,10,10)
 No Repetition: such as lottery numbers (2,14,15,27,30,33)

16. 1. Combination with Repetition
where n is the number of things to choose from, and we choose r of them
(Repetition allowed, order doesn't matter)

17. COMBINATION EXAMPLES:
1. Combination without repetition: Coleen is on a shopping spree. She
buys six tops, three shorts and 4 pairs of sandals. How many different
outfits consisting of a top, shorts and sandals can she create from her
new purchases?
Using this formula:
6C1* 3C1* 4C1 = (6)(3)(4) = 72 possible outfits

18. 2. There are 12 boys and 14 girls in Mrs. Schultzkie's math class. Find the
number of ways Mrs. Schultzkie can select a team of 3 students from the class
to work on a group project. The team is to consist of 1 girl and 2 boys.
Order, or position, is not important. Using the multiplication counting
principle,
3. A box contains 4 red, 3 white and 2 blue balls. Three balls are drawn at
random. Find out the number of ways of selecting the balls of different colours?
1 red ball can be selected in 4C1 ways
1 white ball can be selected in 3C1 ways
1 blue ball can be selected in 2C1 ways
Using calculator: 4C1* 3C1* 2C1 or C(4,1) x C(3,1) x C(2,1) = 24 ways

19.  http://www.pballew.net/permute.html
 http://www.britannica.com/topic/permutation
 http://www.regentsprep.org/regents/math/algtrig/ats5/lcomb.htm
 http://www.cubers.com/classroom/lesson1-2.php
 https://cbpowell.wordpress.com/2009/11/14/permutations-vs-
combinations-how-to-calculate-arrangements/
 https://www.mathsisfun.com/combinatorics/combinations-
permutations.html