The fundamental principle of counting states that if one event can occur in p ways and another independent event can occur in q ways, then both events together can occur in p × q ways.
Example: If you have 3 shirts and 4 pants, you can make 3 × 4 = 12 different outfits.
For sequential choices, multiply the number of options at each step. For alternative choices (either/or), add the number of options.
Permutation: Arrangement of objects where ORDER matters.
Combination: Selection of objects where order does NOT matter.
Probability: Numerical measure of the likelihood that an event will occur.
Remember: Permutations = Positions matter, Combinations = Committees don’t care about order!
For a positive integer n, the factorial of n (denoted as n!) is the product of all positive integers less than or equal to n.
Example: 5! = 5 × 4 × 3 × 2 × 1 = 120
Important Properties:
When simplifying factorial expressions, expand until you have common factors that can be canceled.
Example: 8!/6! = (8×7×6!)/6! = 8×7 = 56
Simplifying factorial expressions:
Example 1: 7!/5! = (7×6×5!)/5! = 7×6 = 42
Example 2: 10!/(3! × 7!) = C(10, 3) = 120
Important identities:
Remember: 0! = 1 (not 0!). This is a convention that makes many formulas work correctly.
A permutation is an arrangement of objects in a specific order. The number of permutations of n distinct objects taken r at a time is:
Example: Number of ways to arrange 3 books from a set of 5 on a shelf:
P(5, 3) = 5!/(5-3)! = 5!/2! = 120/2 = 60 ways
Special Cases:
If the problem mentions “arrangements”, “order”, “positions”, or “sequences”, it’s likely a permutation problem.
1. Permutations with Repeated Objects:
If there are n objects with n₁ alike of one kind, n₂ alike of second kind, …, nₖ alike of kth kind, then:
Example: Arrangements of “WAPDA”:
Total letters = 5, A appears twice
Number of arrangements = 5!/2! = 120/2 = 60
2. Circular Permutations:
Number of ways to arrange n distinct objects in a circle:
3. Necklace/Keyring Permutations:
For arrangements that can be flipped (like necklaces):
For circular arrangements: “Take one seat to fix position, then arrange the rest.” That’s why it’s (n-1)! instead of n!.
A combination is a selection of objects where order does NOT matter. The number of combinations of n distinct objects taken r at a time is:
Example: Number of ways to choose 3 students from a class of 10 for a committee:
C(10, 3) = 10!/(3! × 7!) = 120 ways
Also denoted as: ⁿCᵣ, C(n, r), or binomial coefficient (n choose r)
If the problem mentions “committees”, “teams”, “selections”, “groups”, or “combinations”, it’s likely a combination problem.
Key Properties:
Restricted Combinations:
Geometry Applications:
Pascal’s Triangle shows combination values: Each entry is the sum of the two above it.
Probability is a numerical measure of the chance that a particular event will occur.
Key Terms:
Example: Tossing a fair coin
Sample space S = {H, T}
Event E = getting head = {H}
P(E) = 1/2 = 0.5
Probability always lies between 0 and 1 inclusive: 0 ≤ P(E) ≤ 1
P(impossible event) = 0, P(certain event) = 1
Types of Events:
Probability Rules:
For mutually exclusive events, use ADDITION. For independent events, use MULTIPLICATION.