Sets, functions and counting
How many 10 diget license plates can you make with no repetition from the numbers {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}?
10!
How many ways can you make a 10 diget license plate from the numbers {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}?
10^10
How many ways can you make a 6 diget license plate from the numbers {0, 1, 2, 3, 4, 5, 6, 7 ,8, 9}?
10^6
A coach must choose five starters from a team of 12 players. How many different ways can the coach choose the starters?
12C5
How many 5 diget numbers with exactly one 3?
1x9x9x9x9=9^4=6,561
How many 6 diget license plates have exactly one 4? With repetition and range from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
1x9x9x9x9x9=9^5=59,049 multiply by 6 places=59,049x6=354,294
In how many ways can 3 different vases be arranged on a tray?
3!=3x2x1=6
How many ways to get a straight hand (same suit in order)?
4 suits 10 cards choose a suit =4C1 choose 5 cards from that suit 10C5
How many flush suits (all the same suit)?
4 suits 13 cards in each suit choose a suit 4C1 then choose 5 cards from that suit 13C5 4C1x13C5
How many different 4 letter words can be formed from the letters in the word MATH ?
4x3x2x1=24
How many 5 card poker hands are there?
52C5 = 2,598,960
If you have 6 distinct apples how many ways can you put them in 5 boxes?
5x5x5x5x5x5=5^6
How many 5 diget numbers have no 3's?
8x9x9x9x9=8x9^4=52,488
How many 5 diget numbers are there?
9x10x10x10x10=90,000
How many 5 diget numbers with at least one 3?
9x10x10x10x10=90,000 8x9x9x9x9=52,488 90,000-52,488=37,512
Domain
Set of all first coordinates
Codomain
Set of all possible outcomes
Range
Set of all the actual outcomes
How many poker hands with at least one 3?
count ones with no 3's= 48C5 total amount minus 48C5= 52C5-48C5 =886,656
nC1
n
nP1
n
nPn
n!
Combination Formula
nCr=n!/r!(n-r)!
Permutation Formula
nPr=n!/r!
How many hands with exactly 3?
out of 4 (3's) choose 1= 4C1 for the rest choose 4 out of 48 =48C4 multiply 4C1x48C4
How many hands with exactly two 3's?
out of 4(3's) choose 2=4C2 for the rest choose 3 out of 48 = 48C3 4C2 x 48C3
Why do we study functions?
to analyze the relationships between functions
How many domains are in a function?
1
nCn
1
There are fourteen juniors and twenty-three seniors in the Service Club. The club is to send four representatives to the State Conference. If the members of the club decide to send two juniors and two seniors, how many different groupings are possible?
Choose 2 juniors and 2 seniors 14C2x23C2
There are fourteen juniors and twenty-three seniors in the Service Club. The club is to send four representatives to the State Conference. How many different ways are there to select a group of four students to attend the conference?
Choose 4 students from the total number of students. Order is not important. 37C4
Selecting three students to attend a conference in Washington, D.C. permutation or combination?
Combination, order doesn't matter
A teacher is making a multiple choice quiz. She wants to give each student the same questions, but have each student's questions appear in a different order. If there are twenty-seven students in the class, what is the least number of questions the quiz must contain?
If there were two questions on the quiz, we could prepare two quizzes with the questions in different order -- 2•1 = 2. If there were three questions, we could get 3•2•1 = 6 different orders. If there were four questions, we could get 4•3•2•1 = 24 different orders -- not quite enough for the class of 27 students. If there were five questions, we could get 5•4•3•2•1 = 120 different orders. The teacher will need at least 5 questions on the quiz.
Permutation
Order DOES matter
Assigning students to their seats on the first day of school. Combination or permutation?
Permutation
Selecting a lead and an understudy for a school play. Combination or permutation?
Permutation
Function
Relationship between two quantities First coordinate appears only once Every first element is assigned a second