Permutations and Combinations Basics

Permutations and Combinations Formulas

• Permutations: Arrange n distinct objects in r positions. Formula: P(n, r) = n! / (n-r)! Use when order matters.

• Combinations: Choose r objects from a set of n distinct objects. Formula: C(n, r) = n! / (r! * (n-r)!) Use when order does not matter.

Basic Probability Concepts

• Probability of an event: A numerical measure of the likelihood of an event occurring. Value ranges from 0 (impossible) to 1 (certain).

• Complement of an event: The probability of an event not occurring is 1 minus the probability of it occurring.

Simple Examples

• Arranging 6 books: 6! = 720 ways

• Choosing 4 members from 12: C(12, 4) = 495 ways

• Rolling a 5 on a fair die: 1/6

• Flipping heads: 1/2

• Probability of an event not occurring (0.25 probability): 1 - 0.25 = 0.75

• Drawing an ace: 4/52 = 1/13

• Arranging letters in "MATH": 4! = 24 ways

• Probability of a certain event: 1

Situational Permutations and Combinations

• Committee of 3 from 10: C(10, 3) = 120 ways

• 3-digit code from 0-9 (no repeats): P(10, 3) = 720 possible codes

• Top 3 finishers in 8-runner race: P(8, 3) = 336 ways

• 5 people seated in a row, 2 together: (4!) * 2! = 48 ways

• President and Vice-President from 6 candidates: P(6, 2) = 30 ways

• 4-digit code with repeats allowed: 10 * 10 * 10 * 10 = 10,000 ways

• 5 students arranged in a line: 5! = 120 ways

• 2-person leadership team from 7: C(7, 2) = 21 ways

• Answering 8 multiple-choice questions (4 choices each): 4^8 = 65,536 ways

• Choosing 3 students from 20: C(20, 3) = 1140 ways

Probability with Dice, Coins, and Cards

• Sum of 7 when rolling two dice: 6/36 = 1/6

• Possible sequences of heads and tails in 4 flips: 2^4 = 16

• Probability of rolling a prime number (2, 3, 5): 3/6= 1/2

• Probability of drawing a red marble: 4 / (4+5+6) = 4/15

• Probability of drawing all 5 hearts: (13/52) * (12/51) * (11/50) * (10/49) * (9/48) = very low

• Probability of at least one heads in two flips: 1 - (1/4) = 3/4

• Probability of at least one 3 when rolling two dice: 1- (5/6 * 5/6) =11/36

• Probability of drawing a king or a heart: (4/52) + (13/52) - (1/52) = 16/52

• Probability of 6 followed by even number: (1/6) * (3/6) = 1/12

• Probability of drawing two face cards: (12/52) * (11/51)

Compound Probability and Dependent Events

• Probability of red ball then blue ball (without replacement): (3/10) * (7/9) = 21/90

• Probability of two spades: (13/52) * (12/51)

• Probability of sum greater than 9 with two dice: (4/36)

• Probability of two non-defective items: (15/20) * (14/19)

• Probability of rolling a 2 or 5: (2/6) = 1/3

• Probability of three 6s in a row: (1/6)^3 = 1/216

• Probability of at least one tails (2 coin flips): 1 - (1/4) = 3/4

• Probability of all three students being boys (20 students total, 10 boys, 10 girls): ((10/20) * (9/19) * (8/18))

• Probability of doubles and heads (2 dice, 1 coin): (6/36) * (1/2) = 1/12

• Probability of two girls from a class of 100 students (60 boys, 40 girls): (40/100) * (39/99)

Distinguishable and Non-Distinguishable Arrangements

• Arrangements of "SUCCESS": 7! / 3! = 840

• Arrangements of "BALLOON": 7! / 2! 2! = 1260

• Arrangements of 5 red & 4 blue balls: (9! / 5! , 4!) = 126 ways

• Arrangements of "PEPPER": 6! / 3! = 120 ways

• 3 red & 4 green balls:(7! / 3! , 4!) = 35 ways

• 7 identical & 3 different books arrangement on a shelf: you can treat 7 identical books as one unique and total arrangements will be (10! / 7!)

• 10 students lined up, 4 twins: treat 4 twins as one person. Total is (7! * 4!)

• "MISSISSIPPI" arrangements: 11! / 4! * 4! * 2! =34,650

• Flag with 3 red, 2 blue, & 2 green stripes: 7! / 3! 2! 2! = 210 ways

• 8 books (3 identical math & 5 different): treat the identical 3 as 1 and use the formula (9! / 3!)