Expert II Permutations And Combinations (12) Flashcards
What is factorial?
A multiplication of the given number by all natural numbers below it.
How do you represent a factorial?
!
What is 6! in expanded notation?
6 x 5 x 4 x 3 x 2 x 1
What is 6! ? Show your work.
6 x 5 x 4 x 3 x 2 x 1 = 30 x 12 x 2
= 30 x 24
= 720
Use a calculator to compute 6! .
On a graphing calculator to get the factorial symbol press three following:
Math -> -> -> 4
6! =720
How can you show any number factorial?
n! is the most common way
What does (n-1)! signify?
Subtract 1 from the number you have decided n is equal to at the moment. Then compute the factorial of one less than that number.
If n = 7 then the expression is equal to 6! = 720
How can you cancel out factorials in expressions that contain n! and n minus any natural number factorialed?
Take the larger number which is n! and change it to n(n-1)! Keep subtracting one to find more factors until the last one matches the other part of that original fraction.
Then cancel out the factorials that match perfectly.
How many permutations can you make on three tiles selected one at a time without replacement? Assume each tile is unique and all tiles must be selected.
3! = 3 x 2 x 1
There are three options for the first round so n = number of options for first round = 3
There are three rounds so r = 3
The formula for a permutation is n! / (n - r) !
Since we are sampling until the end and all tiles must be used the denominator is equal to (3-3)! = 0! = 1 and this does not affect the result.
So in this case you can use n! to find the answer since there are no repeated tiles (they are all unique) and since you are using all tiles (the number of rounds equals the number of options for the first round).
How many permutations can you make using exactly three letters from a seven letter tiles? Assume all letters are unique.
Permutations = n! / (n-r)!
n = number of options for first round = 7
r = rounds = 3
Permutations = 7! / (7 - 3)!
= 7! / 4!
= 7 x 6 x 5
It is important to know every step listed here for future understanding of all types of permutations.
In addition, know how to input this in the calculator:
nPr
Most importantly, recognize that all tiles were unique. If they were not you have to further divide the answer to ensure that you have less responses since GOOD is the same as GOOD, even when the O’s switch places. In that case you need to divide by the factorial of the quantity of that doubled letter. You can do this again for another type of doubled letter. This will also work for combinations later.
What is a permutation?
Where order matters
DOG is different from GOD so they make two options rather than one.
Lock combinations are actually permutations since order matters.
Passwords are permutations.
Ingredients in a cake are combinations but layers of a cake could be considered permutations.
