Counting Flashcards
What are two counting techniques?
1.) Sum rule
2.) Product rule
Mutual exclusive means
Events that can’t be done simultaneoulsy
Sum Rule
If task T₁ and task T₂ can happen in n₁ and n₂ ways respectively and T₁, T₂ are mutually exclusive then any of the task can happen in n₁+n₂ ways.
Product Rule
If task T₁ and task T₂ can happen in n₁ and n₂ ways respectively and T₁, T₂ are mutually exclusive then both or all the tasks can be happened in n₁n₂ ways
remember for sum rule
either
remember for product rule
all tasks or together
Necessary condition for T₁, T₂ in sum rule or product rule
T₁ and T₂ are mutually exclusive
A or B (in terms of counting techniques)
Generally A or B means AUB but if A and B are mutually exclusive then A∩B = Φ, then AUB becomes sum rule
Word for permutation
Arrangement
Permutation
Arrangement of r objects from n distinct objects linearly
Types of permutation and also their formulas
1.) Without repetition
no. of ways = ⁿPᵣ = n! / (n-r)!
2.) With repetition
no. of ways = nʳ
Special case for permutation
In without repetition,
When r = n
then no. of ways = n!
value of 0!
1
value of 1!
1
Permutation with repetitions
Circular Arrangements
Arrangement of n distinct objects at a round table can be done in (n-1)! ways.
Necklace
No. of necklace with n distinct colour beads = (n-1)! / 2
Clockwise / Anticlockwise arrangements
No distinction be made in the clockwise and anti-clockwise arrangements, then person can be seated a round table in (n-1)! / 2 ways
Word for combinations
Selections (unordered selections)
Combinations
Permutation in terms of selection
Ordered selection
ⁿcᵣ (proper formula)
ⁿcᵣ = n! / r! (n-r)!
ⁿcᵣ (short)
ⁿcₙ₋ᵣ
Combinations with repetitions
Another statements used for combinations with repetitions
Shortcut for non-empty (Repetitive combination)
Derangement formula
Formula 1 of combination
Formula 2 combination
Formula 3 of combination
Number of diagonals in polygon
Explain whole concept
Total number of different lines from n points where p points are collinear
