Combinatorics
The different combinations from specific finite sets
It has some restrictions
1. Repetition
2. Order
3. Different Criteria
So we have 3 things
1. Permutation
2. Variations
3. Combinations
# No of favorable outcomes
# No of all elements in a sample space
Permutations
The number of different possible ways we can arrange a set of elements
P n = n x (n-1) x (n-2)x............ 1 = n!
n! = Product of numbers from 1 to n
Negative numbers don't have factorial
0!=1
Example
6! = 5! x 6
7! = 6! x 7
(n + k)! and (n - k )!
(n + k)!= n! x ( n+1) x (n+2) x .......(n+k)
(n - k)!= n!
(n - k + 1) x ( n - k + 2)..........x n
n > k
n!/k!= ( k + 1) x ( k + 2) x ............... x n
7!/4!= 5 x 6 x 7
Variations
The total number of ways we can pick and arrange some elements of a given set
The intuition behind the formula. (With Repetition)
• We have n - many options for the first element.
• We still have n - many options for the second
element because repetition is allowed.
• We have n- many options for each of the p- many elements
Variations Don't take into account double counting element
The intuition behind the formula. (Without Repetition)
• We have n- many options for the first element.
• We only have (n-1)-many options for the second
element because we cannot repeat the value for we
chose to start with.
• We have fewer options left for each additional element.
The number of different ways we can pick certain elements of a set
* All the different permutations of a single combination are different variations
C= V/P
A B C
B A C
C B A -- These are different Variation, not different Combination
B C A
C A B
A C B
Let us have to choose 3 names out of 10 name
for 1 combination we have 6 Variation
V(10,3)= 10 x 9 x 8 = 720
C(10,3)= 720/6= 120
** Symmetry of Combinations
Picking more elements can lead to having fewer combinations
since selecting p elements is the same as omitting n-p elements
A Combination can be a mixture of different smaller individual events
**Combinations with separate sample spaces
Combinations represent the number of different possible ways we can pick a number of elements
Characteristics of Combinations with separate sample spaces:
• The option we choose for any element does not affect the number of options for the other elements.
• The order in which we pick the individual elements is arbitrary.
• We need to know the size of the sample space for each individual element. (𝑛1, 𝑛2 … 𝑛𝑝)
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