Probability measures the likelihood of an event, while permutations and combinations count possible arrangements and selections. These topics are closely connected in questions involving cards, dice, teams, arrangements, and selections.

Factorials

Factorial notation is used in counting formulas. For a positive integer n, n! = n x (n - 1) x ... x 2 x 1. Also, 0! = 1.

• 5! = 5 x 4 x 3 x 2 x 1 = 120.
• Cancel factorial terms before expanding them fully whenever possible.

Permutations

A permutation is an arrangement in which order matters. The number of ways to arrange r objects selected from n distinct objects is nPr = n! / (n - r)!.

• Arrange all n distinct objects in n! ways.
• When objects repeat, divide by the factorial of each repeated group.
• For circular arrangements of n distinct objects, the number of arrangements is (n - 1)!.
• Use permutations for rankings, seating orders, and forming ordered numbers.

Combinations

A combination is a selection in which order does not matter. The number of ways to choose r objects from n distinct objects is nCr = n! / [r! x (n - r)!].

• nCr = nC(n - r), which can simplify calculations.
• Use combinations for teams, committees, and groups.
• The relation nPr = nCr x r! connects arrangements and selections.

Basic probability

For equally likely outcomes, probability of an event E is favorable outcomes / total possible outcomes. A probability is always between 0 and 1 inclusive.

• An impossible event has probability 0; a certain event has probability 1.
• P(not E) = 1 - P(E).
• Convert probability to a percentage by multiplying by 100.

Combined events

• Mutually exclusive events: P(A or B) = P(A) + P(B).
• General addition rule: P(A or B) = P(A) + P(B) - P(A and B).
• Independent events: P(A and B) = P(A) x P(B).
• Conditional probability: P(A given B) = P(A and B) / P(B), when P(B) is not zero.
• For selections without replacement, update the total and favorable outcomes after every draw.

How to solve questions

1. Decide whether the problem asks for an arrangement, a selection, or a probability.
2. Use permutations when order matters and combinations when it does not.
3. List the total equally likely outcomes before counting favorable outcomes.
4. Check whether events are mutually exclusive, independent, or conditional.
5. For restricted arrangements, count the complement or use cases when that is simpler.