Combinatorics Flashcards
What does combinatorics deals with?
combinations of objects from a specific, finite set
What Restrictions can be formed to apply to combinations?
Repetition
Order
Or other criterion
What are the 3 integral parts of Combinatorics?
- Permutations
- Variations
- Combinations
we use these to determine the number of favorable outcomes OR
the number of all elements in a sample space
What do Permutations represent?
Permutations represent the number of different possible ways we can arrange a set of elements
What is the formula for Permutations?
n!
How are Permutations denoted?
P()
Going back to the Formula 1 example from the lecture, in how many ways can the 20 drivers finish a race?
20!
When we determine the available options for every position in a permutation, can we start with the middle element?
Yes, choosing the order is completely up to us.
What are factorials denoted as?
!
What does n! represent?
The product of the natural numbers from 1 to n
ie. n! = 1 x 2 x 3 x …. x n
3! = 1 x 2 x 3 = 6
What is one odd characteristic of factorials?
Negative numbers don’t have a factorial
What are some important properties of factorials?
n! = (n-1)! x n
(n + 1)1 = n! x (n +1)
What is 5! Equal to?
120
Which of the values below is NOT the equivalent to “6!” ?
A) 123456
B) 7!/7
C) 4!5*6
D) 8!/8
8!/8
Which of the following is equivalent to 5!/3! ?
A) 45/6
B) 123
C) 45
D) 1*2
C) 4*5
What is the formula for Variations without Repetition?
V = n! / (n - p)!
The number of variations without repetition when arranging ‘p’ elements out of a total of ‘n’ elements.
What do Variations express?
The total number of ways we can pick and arrange some elements of a given set
What is ‘n’ in probability theory?
n = the total number of elements we have available
What is ‘p’ ?
p = the number of positions we need to fill
When arranging ‘p’ elements out of a total of ‘n’
What is the Notation and Formula for calculating Variations with Repetition?
V = n to the power of p
V bar, n, p = n to the p
n = the total number of elements we have available
p = the number of positions we need to fill
The number of variations with repetition when picking p-many elements out of n elements, is equal to n to the power of p
What is the formula for Variations with Repeating values?
n to the power of p
When do we use Variations instead of Permutations?
When we are not arranging all elements in the sample space.
Going back to the combinations lock example from last lecture, imagine the correct code consisted of a single 4-letter word. Remember that we can only use the letters A, B and C. How many possible passcodes are there?
3^4
4^3
2^3
26^4
n to p
or
3^4
How do we calculate Variations for events without repetition?
We cannot use the same element twice
example element: person in running schedule