4.3 Counting Rules
In how many ways can the letters in the word 'Missouri' be arranged?
8!÷(2!*2!) 8 total letters ÷ by the repeats
You are getting a line-up ready for a school kickball game. You have 7 girls and 7 boys. The rules state each child must kick the same number of times and alternate girl-boy or boy-girl. How many ways can a line-up be made for one round of kicking?
First we will begin with the girl-boy line-up. 7!⋅7!=25,401,600 Now we will look at the boy-girl line-up. (7!⋅7!)⋅2=25,401,600⋅2=50,803,200
You need to have a password with 4 letters followed by 3 even digits between 0 and 9, inclusive. If the characters and digits cannot be used more than once, how many choices do you have for your password?
For this type of problem, we will use the Permutation Rule. 5 numbers b/c odd = 1,3,5,7,9 ₂₆P₄*₅P₃
Example of combination:
However, if you were trying to choose 33 runners to race, the team of Chelsie, Chris, and Katie would be the same team as Chris, Katie, and Chelsie. In this case we would say that order is not important. When the order in which the objects are chosen is not important, then we use a combination.
Scramble Mississippi: how many unique combinations can you make
Mississippi: some letters repeat begin with 11 and then ÷ i repeats 4x in a group of 11 letters s repeats 4x in a group of 11 letters p repeats 2x in a group of 11 letters 11!÷(4!*4!*2!) <-- put into calculator just like this =34,650 unique combinations
How many different ways can the letters in TEXAS be arranged to form five letter "words"?
Solution: In this problem, we are not concerned with forming actual words but are simply trying to determine how many different ways the five letters can be arranged. The first task is to choose which letter (t, e, x, or S) will come first. There are five letters from which to choose, so the first experiment has five outcomes. Once this first task has been completed and the first letter chosen, we move on to the next task of choosing the second letter. There are now only four letters from which to choose, and so there are only four outcomes for the second experiment. Similarly, there will be only three letters from which to choose for the third task, and so on. Using the Fundamental Counting Principle, it is then easy to see that the number of arrangements of TEXAS is 5⋅4⋅3⋅2⋅1 = 120
an example of the fundamental principle of counting
Suppose a restaurant offers a value meal consisting of a sandwich and a drink. If there are 5 types of sandwiches and 4 types of drinks, how many different meals are possible? In this problem there are two tasks to be performed, namely, choosing a sandwich and choosing a drink. For each sandwich choice, there are 4 possibilities for the drink choice. The end result is that there are 5⋅4=20 total possible meals or outcomes.
For a committee of 8 people, how many ways can a chairperson and a secretary be selected from amongst its members? what is n and what is r?
The sample size is 8, so n=8. Since two positions are to be filled, r=2.
A value meal package at Ron's Subs consists of a drink, a sandwich, and a bag of chips. There are 6 types of drinks to choose from, 4 types of sandwiches, and 3 types of chips. How many different value meal packages are possible?
Use The Fundamental Principle of Counting. 6x4x3
7 cards are drawn from a standard deck of 52 playing cards. How many different 7-card hands are possible if the drawing is done without replacement?
Use the Combination Rule. 52!÷7!*(52−7)!
How many ways can Rudy choose 5 pizza toppings from a menu of 19 toppings if each topping can only be chosen once?
Use the Combination Rule. ₁₉C₅
A person tosses a coin 9 times. In how many ways can he get 6 heads?
Use the Combination Rule. ₉C₆
combination
When the order in which the objects are chosen is not important
How many ways can four people be chosen from a group of twenty to serve on a committee?
order is not important for committee n=20 r=4 in your calculator: type 20, then press math scroll to PRB & scroll down to 2:nCr - press enter type 4 - press enter =4845
For a committee of 8 people, how many ways can a chairperson and a secretary be selected from amongst its members? Is this a permutaion or a combination?
order of the members chosen is important, i.e., it is different if someone is elected for chairperson rather than secretary. Therefore, this is a permutation.
typically you will see more _____ than _____
permutation than combination
Permutation rule
read "n things permuted r at a time,"
Combination rule
read in the shorter form "n choose r."
Special Permutations
some of the objects being counted are identical. For example, consider the permutation on the letters in the word Mississippi. Because the letters "i", "s", and "p" are repeated we must use a different permutation formula which takes into consideration the repeated objects, or in this case, the repeated letters.
There are 12 people in an office with 6 different phone lines. If all the lines begin to ring at once, how many groups of 6 people can answer these lines?
the given problem is a combination. 12!÷6!(12−6)!
Factorials
mathematical expression used in many probability calculations A factorial is the product of all positive integers less than or equal to n. Symbolically, factorials are written as n!=n(n−1)(n−2)...(2)(1) By definition, 0!=1.
Permutation example:
suppose that a group of ten runners is racing. If the first three runners finish in the order: Chelsie, Chris, then Katie, then this is different than them finishing Chris, Katie, then Chelsie, especially to Chelsie! Although they are the same three people, these are two different outcomes for the race.
to use factorials on the TI84
67!÷65! enter 67 and press math scroll right to PRB then down to 4:! and press enter press ÷ enter 65 and repeat the steps through enter
remember, the denominator must only consist of multiplication or addition, not both:
a combination of, example: turn the (6-2) into a 4! and then put it into the calculator : 8!÷(4!*4!)
Fundamental Principle of Counting
It states that you can multiply together the number of possible outcomes for each stage in an experiment in order to obtain the total number of outcomes for that experiment.
Solve: For a committee of 8 people, how many ways can a chairperson and a secretary be selected from amongst its members?
in your calculator: type 8, then press math scroll to PRB & scroll down to 2:nPr - press enter type 2 - press enter =56
How many different car license plates are possible if a license plate consists of three numbers followed by three letters of the alphabet?
Solution: There are 6 slots to fill - 3 are for the numbers and 3 are for the letters. Each of the first 3 slots has 10 different choices (the digits 0-9). Each of the last 3 slots has 26 different choices (the 26 letters in the alphabet). Thus, by the Fundamental Counting Principle the total number of plates possible is 10⋅10⋅10⋅26⋅26⋅26 = 17,576,000
A coordinator will select 7 songs from a list of 9 songs to compose an event's musical entertainment lineup. How many different lineups are possible?
We can use the formula for the permutation of n elements taken r at a time. ₉P₇
A doctor visits her patients during morning rounds. In how many ways can the doctor visit 8 patients during the morning rounds?
We can use the formula for the permutation of n unique elements. There are 8 patients or "objects" being permuted. So, the number of ways the doctor can visit the 8 patients is 8!=40320
When you are looking to count the number of ways objects can be chosen out of a group, then the problem you are dealing with is either
a permutation or a combination.
Since factorials equal the product of a string of positive numbers, their values get
very big, very quickly 5×4×3×2×1=120, could have been written more easily as 5!=120 While it was easy to calculate 5! without a calculator, expressions such as 100! are much too large even for ordinary calculators to compute. Therefore, we will only look at relatively small values of n!
A helpful method for setting up a Fundamental Counting Principle problem is to think of each stage as
a "slot" First, determine the number of slots that must be filled. Then, decide how many outcomes are possible for each slot. For example, suppose one university issues student identification numbers that consist of two letters followed by three digits (0-9). How can we determine how many unique ID numbers the school can assign? The first step is to notice that there are 5 ID number "slots" to be filled. The first two slots contain 26 outcomes each, one outcome for each letter of the alphabet. The other 3 slots contain 10 possibilities each, one for each digit 0-9. Finally, we simply multiply the number of possibilities for each slot together and obtain 26×26×10×10×10 = 676,000 unique ID numbers.
Once you have determined whether the order in which objects are chosen is important in your problem, you
apply one of the rules to calculate the total number of possibilities
You are ordering a hamburger and can get up to 7 toppings, but each topping can only be used once. You tell the cashier to surprise you with the toppings you get. What is the probability that you get 1 topping? Express your answer as a fraction or a decimal number rounded to four decimal places.
choosing 1 topping out of 7 is a combination. nCr = n!÷r!(n−r)! ₇C₁
permutation
the order in which the objects are chosen is important That is, the members of the group are picked out in a particular order.
