Basic counting principles, day 1 To decide how many ways something can occur, you can draw a tree diagram. Note that this only shows half of the tree – the automatic transmission

Possible choices = 2 * 3 * 4 = 24 choices transmission * music * color Possible choices = 2 * 3 * 4 = 24 choices transmission * music * color Basic counting principle: If an event can occur in p ways, and another event in q ways, then there are p * q ways both events can occur. Basic counting principle: If an event can occur in p ways, and another event in q ways, then there are p * q ways both events can occur. From now on we will use multiplication and “fill the slots” as follows. From now on we will use multiplication and “fill the slots” as follows.

How many different batting orders are there in a 9- person softball team? Fill the slots for each position 1 st batter 2 nd batter 3 rd batter 4 th batter 5 th batter 6 th batter 7 th batter 8 th batter 9 th batter

9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 9 ! 1 st batter 2 nd batter 3 rd batter 4 th batter 5 batter 6 th batter 7 th batter 8 th batter 9 th batter 9 ways to choose first batter 8 ways left, once 1st batter chosen 7 ways now to chose 3rd batter Etc. 9! = 362,880 ways to write the batting order.

How many 7 digit phone numbers are there if the first digit cannot be 0 or 1? ____*____*___* ____*____*____*____ ____*____*___* ____*____*____*____

8*10* * * * * # choices for first digit # choices for second digit # choices for all other digits will be the same 8,000,000 ways

How many 7-digit phone numbers begin with 867? ____*____ *____*____*____*____*____ ____*____ *____*____*____*____*____

one choice these four digits can be 0-9 (10 choices) 1 * 1 * 1 * 10 * 10 * 10 * 10 10,000 different phone numbers

Using letters from the word “MATRIX” How many 4-letter patterns can be formed? How many 4-letter patterns can be formed? # ways to fill slots: _____ * _____ * _____ * _____ _____ * _____ * _____ * _____

6 letters to choose from 5 letters left.. etc. # ways to fill slots: 6 * 5 * 4 * 3 = 360

Still using letters from the word “MATRIX”, what if the first letter must be a vowel? _____ * _____ * _____ * _____ _____ * _____ * _____ * _____

2 vowels to choose from 5 letters left.. etc. # ways to fill slots: 2 * 5 * 4 * 3 = 120

What if we fill only 4 slots and the first and last letters must be consonants? ______ * ______ * ______ * ______

4 conso- nents to choose from 4 letters left.. 3 conso- nents left to choose from 3 letters left.. # ways to fill slots: 4 * 4 * 3 * 3 = 144 fill these two first fill less important slots last

How many 3-digit palindromes are possible? Note that a number does not usually begin with a zero... ______ * ______ * ______

Can’t begin with a zero and be 3- digit must be the same as the first digit (one way to fill) can be any digit 9 * 10 * 1 = 90 fill first fill second then fill the last one

In a 5-card poker hand, the 1 st 3 cards were red face cards, the last 2 were black non- face cards. How many ways can this happen? 5 slots to fill: _____ * ____ * _____ * _____* _____ _____ * ____ * _____ * _____* _____

first 3 cards choices: K, Q, J hearts or diamonds; 1 st slot, 2 nd slot, 3 rd slot A – 10 and black: 20 cards; 1 st slot, 2 nd slot (of that type) 6 * 5 * 4 * 20 * 19 = 45,600 ways

Mutually Exclusive events To get to school, Rita can either walk or ride the bus. If she walks, she can take 3 routes, if she rides the bus, there are 2 routes. Rita’s choices are mutually exclusive - she can’t do both, therefore, the possibilities are added to each other. To get to school, Rita can either walk or ride the bus. If she walks, she can take 3 routes, if she rides the bus, there are 2 routes. Rita’s choices are mutually exclusive - she can’t do both, therefore, the possibilities are added to each other. 3 ways to walk 3 ways to walk 2 ways to take a bus 2 ways to take a bus total possible routes: = 5 total possible routes: = 5

How many odd numbers between 10 and 1000 start and end with the same digit? We have two mutually exclusive events: ___ * ___ two digit choices + ___ * ___ * ___ three digit choices

two digit choices: just count multiples of 11 that are odd: (11, 33, 55, 77, 99) 5 choices three digit choices: calculate like our palindrome problem (3 slots to fill) 1 * 10 * 5 50 choices total = = 55 odd numbers between 10 and 1000 that have the same first and last digit.

An example that is not mutually exclusive… (each time you have the same number of choices) An ID label has 4 letters. How many different labels are possible? ____ * _____ * _____ * _____ ____ * _____ * _____ * _____

From before, we know it is From before, we know it is 26* 26 * 26 *26= 26 4 = 456,976 26* 26 * 26 *26= 26 4 = 456,976 You can also look at it like: You can also look at it like: 26 4 where 26 (base) = number of different choices 4 (exponent) = number of times you make that choice

On a multiple choice test with 15 questions and 4 answer choices per question, how many answer combinations are there? On a multiple choice test with 15 questions and 4 answer choices per question, how many answer combinations are there? 4 15 =1,073,741,824!!!! 4 15 =1,073,741,824!!!! 4 = number of different choices 15 = number of times you make that choice

How many license plates of 2 symbols (letters and digits) can be made using at least one letter in each? How many license plates of 2 symbols (letters and digits) can be made using at least one letter in each? cases: cases: one letter: L D or D L 2(26*10) one letter: L D or D L 2(26*10) two letters: two letters: L L 26 2 L L 26 2 total: 2(26*10) = 1196 license plates total: 2(26*10) = 1196 license plates

How many ways can you make a 3 symbol license plate using at least one letter? What are the cases? What are the cases?

Cases if the license plate has at least one letter: Cases if the license plate has at least one letter: one letter: one letter: LDD or DLD or DDL3(26*10 2 ) LDD or DLD or DDL3(26*10 2 ) two letters: + two letters: + L L D or L D L or D L L 3(26 2 *10) L L D or L D L or D L L 3(26 2 *10) three letters: + three letters: + L L L 26 3 L L L 26 3

Permutations A permutation is an ordered arrangement: ORDER MATTERS (KAT is different than TAK). Our batting order problem and use of the letters of matrix have been permutations. A permutation is an ordered arrangement: ORDER MATTERS (KAT is different than TAK). Our batting order problem and use of the letters of matrix have been permutations.

How many ways can letters in SPRING be arranged? How many ways can letters in SPRING be arranged? 6 * 5 * 4 * 3 * 2 * 1 = 6! = 720 ways to arrange the 6 letters when order matters. 6 * 5 * 4 * 3 * 2 * 1 = 6! = 720 ways to arrange the 6 letters when order matters. In general, there are n! permutations of n objects. In general, there are n! permutations of n objects.

What if we want to use the letters in SPRING, but only want to know how many 2 letter arrangements can be made? What if we want to use the letters in SPRING, but only want to know how many 2 letter arrangements can be made? Using the “filling the slots” idea, we have Using the “filling the slots” idea, we have 6 * 5 = 30 ways to fill two slots with 6 letter choices. 6 * 5 = 30 ways to fill two slots with 6 letter choices.

Another way to look at it is similar to what we did for Pascal’s triangle: In general, the number of permutations of n objects taken r at a time is:

From 1000 contest entries, how many ways can 1st, 2nd, and 3rd place prizes be awarded? Does order matter? We can fill in the slots: _____ * _____ * _____ _____ * _____ * _____ Or use our formula:

How does it change if the values are not all unique? How many 6-letter permutations of ACOSTA are there? note: there are two A’s that will not be distinguishable from each other in a word. A 1 A 2 COST = A 2 A 1 COST the number of permutations will be cut in half.

Is the bottom just “2” because there are 2 A’s? What if there were more? Consider using the letters of PARABOLA:

Consider PARABOLA the number of arrangements of the 3 A’s that will be equivalent are: A 1 A 2 A 3 PRBOL A 1 A 3 A 2 PRBOL A 2 A 1 A 3 PRBOL A 2 A 1 A 3 PRBOL A 2 A 3 A 1 PRBOL A 3 A 1 A 2 PRBOL A 3 A 2 A 1 PRBOL This shows that 3! arrangements would be identical (the number of ways you can arrange 3 objects in different order). Thus, we need to divide by 3!, not just 3.

This generalizes to objects that have more than one duplicate. If all objects are used, the number of permutations of n objects of which p and q represent the number of items that are alike is:

How many permutations of the letters in the word POSSIBILITY using only 5 letters are there?

where the denominator represents the 2 S’s and the 3 I’s.

That’s all folks Have a counting, permutable kind of day