U4 Flashcards
How to use the fundamental counting principle
multiply the number of choices for each stage together
Before finding the number of options for other stages what do you HAVE to do first
Consider any restrictions for a particular stage
Does order matter in permutations
YES
What is the notation for a permutation and what do the variables represent
(sub n)P(sub r)
n= total # of objects r= # of objects chosen
What is the purpose of a factorial
A short way to write a multiplication statement of descending WHOLE numbers
What is “n” an element of
Whole numbers
How do you write (4)(3)(2)(1) as a factorial
4!
Is arranging letters of a word a perm or comb
Perm
Is arranging people in a line a perm or comb
Perm
How do you solve
(sub n) P (sub 2)
1) plug into perm formula
2) expand n! on the top to
n(n-1)(n-2)!
3) cancel the (n-2)! on the bottom and top
4) foils out and factor as a quadratic
Can “n” ever be a negative #
And why
Can’t have a negative amount of objects to pick from
Permutations: how to answer grouped objects questions
1) ask yourself if you can rearrange the group and write that as a factorial (if you can’t continue as usual)
2) ask yourself if you can move the group and write that as just a # (if you can’t continue as usual)
3) how can you rearrange the rest of the positions and write that as a factorial
4) calculate
What is a compliment
And when do we use it
Outcomes that don’t happen
Use it for “not” questions
A group of 2 people don’t want to sit together, what would be the compliment
They do sit together
What is the compliment formula
Not on sheet
Compliment=
(total with no restrictions)-(outcomes that do occur)
Both are written as factorials
What is the formula for permutations with repeated objects
And what do the variables represent
n!/(a! b! c!)
Not on formula sheet
n= # of total objects
a b c= the same of one type (so don’t have to use all of them)
How is VANCOUVER written in the perm formula for repeated objects
9!/ 2!
What formula do square path questions use
Formula for permutations with repeated objects
How to solve square path questions
1) count the number is increments vertical and count the number of increments horizontal
2) these two numbers are the bottom values in the reputation formula
3) add these two numbers to get the top value (n) in the report on formula
4) plug numbers in and calculate
What is every value in the repetition formula written as
A factorial
Why doesn’t the order matter in combinations
Because it creates the exact same group
What are 3 examples of combinations
- committee where no roles are being served
- card hands
- lottery tickets
What does “or mean”
Add
What does “and” mean
Multiply
How can you use “and” and “or” to help answer questions
Write out what is being asked in words using those words
Cards: how many cards does each suit have
13
How to solve combination problems involving “at least” or “at most” scenarios
1) find the # of outcomes for each individual scenario
2) add them up
What is the first thing you must always do when solving a perm or comb question
Determine if it’s a perm or comb
Can a question have a perm component and comb component
Yes
Concept 4 example 3
Fill in
Properties of binomial Theorem: there are _____ terms in the expansion
n+ 1
Properties of binomial Theorem: the sum of the exponents x and y in EACH term is equal to _____
(And what does this look like)
_____
if n=5, x^3y^2
Properties of binomial Theorem: the exponents of x ______ term by term from n to 0
Decrease
Properties of binomial Theorem: the exponents of y ______ term by term from 0 to n
Increase
Properties of binomial Theorem: the coefficients in each expansion form a symmetrical _____ array
Triangular
In binomial thereom what does n have to be
A whole number
How is pascals triangle limited
Have to write out every row before
General term formula: what does k represent
One less than the term you want
General term formula: what does n represent
The exponent of the binomial
General term formula: what does x represent
First term of the binomial (no expansion)
General term formula: what does y represent
Second term of binomial (not expansion)
Binomial theorem: how do you solve question asking for the middle term
1) must be an odd number of terms on the expansion
2) n+1 = odd number
3) plug values into formula and solve
Binomial theorem: how to solve when either x or y is unknown and so is k
1) find value of k by _____
4) solve for x or y by subbing values into general term formula and simplifying
5) make general term formula equal the term given and solve for x or y by isolating
Binomial theorem: determine the numerical coefficient of the term containing a^7 in the expansion
(3-a)^10
1) plug values into general term formula
2) k=7
3) put everything into general term formula again and solve
Binomial theorem: if asked for the constant term does this mean it’s the last term
No
