Factorials, Permutations and Combinations
Notes found at: http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut55_count.htm
Fundamental Counting Principle: counts the number of ways a task can occur given a series of events. Basically you multiply the number of possibilities each event of the task can occur. Example 1: A deli has a lunch special which consists of a sandwich, soup, dessert and drink for $4.99. They offer the following choices:
Sandwich: chicken salad, ham, and tuna, and roast beef
Soup: tomato, chicken noodle, vegetable
Dessert: cookie and pie
Drink: tea, coffee, coke, diet coke and sprite
How many lunch specials are there? Let’s use the basic counting principle:
There are 4 stages or events: choosing a sandwich, choosing a soup, choosing a dessert and choosing a drink. There are 4 choices for the sandwich, 3 choices for the soup, 2 choices for the dessert and 5 choices for the drink. Putting that all together we get:
Dessert
Soup
Sand.
*
4 * 3 * 2
Drink
5
=
# of lunch specials
120
So there are 120 lunch specials possible.
Example 2: You are taking a test that has five True/False questions. If you answer each question with True or False and leave none of them blank, in how many ways can you answer the whole test?
There are 5 stages or events: question 1, question 2, question 3, question 4, and question 5. There are 2 choices for each question.
quest. 1 quest. 2 quest. 3 quest. 4 quest. 5 # of ways to answer test
2 * 2 * 2 * 2 * 2 = 32
So there are 32 different ways to answer the whole test. 1
Your Turn Example 3: A company places a 6­symbol code on each unit of product. The code consists of 4 digits, the first of which is the number 5, followed by 2 letters, the first of which is NOT a vowel. How many different codes are possible?
There are 6 stages or events: digit 1, digit 2, digit 3, digit 4, letter 1, and letter 2.
In general there are 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The first digit is limited to being the number 5, so there is only one possibility for that one. There are no restriction on digits 2 ­ 4, so each one of those has 10 possibilities.
10
10
10
In general, there are 26 letters in the alphabet. The first letter, cannot be a vowel (a, e, i, o, u), so that means there are 21 possible letters that could go there. The second letter has no restriction, so there are 26 possibilities for that one.
digit 1
digit 2
digit 3
digit 4
letter 1
letter 2
# of codes
= 546000
So there are 546000 different 6­symbol codes possible.
Practice Problems.
1.One quarter, one dime and one six­sided die are tossed. How many results are possible? There are 3 stages or events: one quarter, one dime, and one six­sided die. Each coin has 2 possible outcomes, either a tail or a head. The die has 6 possible outcomes.
quarter
dime
die
# of possible outcomes
2 * 2 * 6 = 24
2. Next semester you are going to take one science class, one math class, one history class and one english class. According to the schedule you have 4 different science classes, 3 different math classes, 2 different history classes, and 3 different English classes to choose from. Assuming no scheduling conflicts, how many different four­course selections can you make?
There are 4 stages or events: a science class, a math class, a history class and an english class.
There are 4 different science classes, 3 different math classes, 2 different history classes, and 3 different English classes to pick from.
Science Math
History English
# of schedules
4 * 3 * 2 * 3
= 72
3. Six students in a speech class all have to give there speech on the same day. One of the students insists on being first. If this student's request is granted, how many different ways are there to schedule the speeches?
There are 6 stages or events: speaker 1, speaker 2, speaker 3, speaker 4, speaker 5, and speaker 6. There is only one possibility for speaker 1. That leaves 5 possibilities for speaker 2, which leaves 4 for speaker 3, which leaves 3 for speaker 4, which leaves 2 for speaker 5 which leaves 1 for speaker 6. speaker 1 speaker 2 speaker 3 speaker 4 speaker 5 speaker 6
# of ways
1 * 5 * 4 * 3 * 2 * 1
= 120 2
Factorial: The factorial function (symbol: !) just means to multiply a series of descending natural numbers.
Ex. 4! = 4*3*2*1 = 7! = 7*6*5*4*3*2*1=
1! = 0! = We needed to know about factorial because it is used the formula for permutation, ­ our next topic.
Permutation
An ORDER of arrangements of r objects, without repetition, selected from n distinct objects is called a permutation of n objects taken r at a time, and is denoted as nPr = n! (n ­ r)!
In other words, when you need to count the number of ways you can arrange items where ORDER is important, then you can use permutation to count. Memory tool ­ Permutations are Persinckety! (picky ­ order matters). A permutation is the choice of r things from a set of n things without replacement and where the order matters.
Remember:
n total
r want(arrangement)
Calculator:
Math key ⇒⇒Prob
#2 nPr permutation
Example: In how many ways can 8 CD’s be arranged on a shelf? Since we are arranging these CD’s that means order is important. First we need to find n and r : n is the number of CD's we have to choose from. What do you think n is in this problem? r is the number of CD’s we are using at a time. What do you think r is?
n = 8 r = 8
nPr = this means there are 40320 different ways to arrange those 8 CD’s
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Example 2: If a softball league has 10 teams, how many different end of the season rankings are possible? (Assume no ties). Since we are ranking these teams that means order is important. So we can use permutations to help us out here.
First we need to find n and r :
n = the number of teams we have to choose from r = the number of teams we are ranking at a time
r = 10
n = 10
Putting this into the permutation formula we get: nPr = there are 3,628,800 different ways to rank those 10 teams
Example 3: In how many ways can a sorority of 20 members select a president, vice president and treasury, assuming that the same person cannot hold more than one office. Since we are choosing offices, which is a way to rank members, that means order is important. So we can use permutations.
First we need to find n and r :
n = the number of members we have to choose from. r = the number of members we are selecting for offices at a time.
n = 20 r = 3
Putting this into the permutation formula we get: nPr = there are 6840 different ways to select the three officers
Example 4: How many different arrangements can be made using two of the letters of the word TEXAS if no letter is to be used more than once? Since we are arranging letters, this means order is important. So we can use permutations First we need to find n and r :
n = the number of letters we have to choose from. r = the number of letters we are using at a time.
n = 5
r = 2
Putting this into the permutation formula we get:
there are 20 different 2 letter arrangements.
Ex. 5: A company issues a questionnaire whereby each employee must rank the 5 items with which he or she is most satisfied. The items are wages, work environment, vacation time, job security, supervisors, health insurance, break time, and retirement plan. The ranking is to be indicated by the numbers 1, 2, 3, 4 and 5, where 1 indicates the item involving the greatest satisfaction and 5 the least. In how many ways can an employee answer this questionnaire? Since we are ranking these items, this means order is important. ­ use permutations.
First we need to find n and r : n = the number of items we have to choose from
r = the number of items we are ranking at a time.
n = 8
r = 5
Now nPr = there are 6720 different ways to rank these items.
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Combination
An arrangement of r objects, WITHOUT regard to ORDER and without repetition, selected from n distinct objects is called a combination of n objects taken r at a time. The number of such combinations is denoted by:
The difference between combinations and permutations is in combinations you are counting groups (order is not important) and in permutations you are counting different ways to arrange items with regard to order.
The n and the r mean the same thing in both the permutation and combinations, but the formula differs. Note that the combination has an extra r! in its denominator. Example 1: In a conference of 9 schools, how many intraconference football games are played during the season if the teams all play each other exactly once?
When the teams play each other, order does not matter, we are counting match ups. For each game there is a group of two teams playing. So we can use combinations
(Note that if we were putting these teams in any kind of order, then we would need to use permutations to solve the problem.)
First we need to find n and r :
If n is the number of teams we have to choose from, what do you think n is in this problem?
If r is the number of teams we are using at a time, what do you think r is?
n = 9 r = 2
there are 36 different games in the conference.
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Example 2: You are going to draw 4 cards from a standard deck of 52 cards. How many different 4 card hands are possible?
This would be a combination problem, because a hand would be a group of cards without regard to order. (Note that if we were putting these cards in any kind of order, then we would need to use permutations to solve the problem.)
First we need to find n and r : If n is the number of cards we have to choose from, what do you think n is in this problem? If r is the number of cards we are using at a time, what do you think r is?
n = 52 r = 4 there are 270,725 different 4 card hands.
Example 3: 3 marbles are drawn at random from a bag containing 3 red and 5 white marbles. 3a. How many different draws are there? ­ what kind of problem is this?
Note that there are no special conditions placed on the marbles that we draw, so this is a straight forward combination problem. This would be a combination problem, because a draw would be a group of marbles without regard to order. It is like grabbing a handful of marbles and looking at them. (Note that if we were putting these marbles in any kind of order, then we would need to use permutations to solve the problem.)
First we need to find n and r:
If n is the number of marbles we have to choose from, what do you think n is in this problem?
If r is the number of marbles we are drawing at a time, what do you think r is?
n = 8 There are 3 red and 5 white marbles for a total of 8 marbles.
r = 3 There are 3 marbles drawn at a time.
there are 56 different draws.
3b. How many different draws would contain only red marbles?
This would be a combination problem, because a draw would be a group of marbles without regard to order. It is like grabbing a handful of marbles and looking at them. In part a above, we looked at all possible draws. From that list we only want the ones that contain only red. Let's see what the draw looks like: we would have to have 3 red marbles to meet this condition: 3 RED
First we need to find n and r :
If n is the number of RED marbles we have to choose from, what do you think n is in this problem?
n = 3 There are a total of 3 red marbles.
If r is the number of RED marbles we are drawing at a time, what do you think r is?
r = 3, 3 RED marbles are drawn at a time.
this means there is only 1 draw out of the 56 found in part a that would contain 3 RED marbles.
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http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut57_comb.htm
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut57_comb.htm
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