Master Permutations, Combinations & Probability Basics

Explore permutation as a counting technique for arranging objects where order matters, learn how to count possible arrangements and number of possibilities, and preview factorial and dnpr concepts.

Explore factorials, their notation, and how to break larger factorials into smaller factors to simplify calculations. Learn cancellation techniques, key properties, and that factorials apply only to whole numbers.

Understand NPR, the permutation count of n distinct items taken r at a time, via n!/(n-r)!, with examples like 6P3 and 7P2, and that nPn equals n!.

Identify and apply the two fundamental counting principles: multiplication for sequential choices and addition for alternatives, with permutation examples illustrating how to count possibilities.

Master counting arrangements in a row using permutations and factorials, with the nPr formula demonstrated through letter-based words and book arrangements.

Explore how to count permutations when items are not all distinct using factorials and division by repeated counts, with examples like apple, Mississippi, independence, and institute.

Learn to count restricted permutations by grouping letters that must be together or apart, using factorials and division for repeats, with examples from independence, Mississippi, and success.

Explore conditional permutations with restrictions by fixing first and last letters, handling repeated letters with factorials, and solving examples like permutations, series, daughter, and subject.

Explore conditional permutations by arranging letters at even or odd places, with vowels occupying specified positions, illustrated using nation, apple, and strange examples such as 18, 1440 and 2880.

Apply the fundamental principle of counting to form 3-, 4-, and 5-digit numbers under distinct-digit and repetition constraints, including zero placement and nPr calculations.

Continue solving number formation problems with different restrictions: form three-digit numbers from digits 1–5 with repetition, and four-digit numbers from 4–8 divisible by five, plus other constrained counts.

Learn how circular permutations reduce to (n-1)!, counting identical arrangements once, and solve seating problems around round tables, including cases with no two women together or two people together.

Tackles challenging permutation problems, including adjacency constraints with letters, counting with the fundamental principle, forming even and divisible numbers under digit restrictions, and arranging five boys and five girls alternately.

Tackle challenging permutation problems, including seating three girls with nine boys in two vans, adjacency constraints in back rows, and ranking words in dictionary order.

Explore two permutation problems: count four-digit numbers from digits 1–4 not exceeding 4321 with repetition, and arrange nine papers so the best and worst never sit together.

Explore how combinations count selections without regard to order, using nCr as n!/(r!(n−r)!), and distinguish them from permutations with examples such as 6C2=15 and 7C3=35.

Explore properties of ncr, including zero and full-set values, symmetry ncr = ncr(n−r), and the identity ncr + ncr−1 = n+1cr.

Practice problems on combinations sharpen skills using nCr to count selections, from 12 choose 2 handshakes to 15 choose 11 team picks and 10 choose 6 questions.

Learn to solve skill problems by combining permutations and combinations to form words from letters, choosing consonants and vowels, and applying factorials to count outcomes.

Practice skilled problems on permutations and combinations, including arranging algebra with vowels and consonants in fixed relative order using factorials, and counting p and s with four letters between them.

Explore skilled permutation and combination problems using word-based examples from Monday and mathematics, applying six P4, four-letter selections, full six-letter permutations, and case analyses of distinct, paired letters.

Explore skilled permutation and combination problems by forming four-letter words from the letters of ineffective, counting all cases to total 1422 possible words.

Explore probability by distinguishing experimental and theoretical approaches, and learn key terms like trial, event, elementary and compound events, determinism, and random experiments with coin and die examples.

Empirical probability is the experimental probability, defined as p(A) equals the number of trials A occurs divided by total trials. Use coin-toss and score examples to illustrate decimals and percentages.

Explore theoretical probability by defining probability as favorable outcomes over total outcomes, with event E and its complement E' summing to one. For a die, P(E)=1/2 and P(E')=1/2.

Explore key probability terms such as sample space, exhaustive events, equally likely events, mutually exclusive events, and odds in favor or against, with total and favorable outcomes.

Explore coin-based probability by counting outcomes for 2 and 3 coins, generalizing to 2^n, and calculating events such as two heads, at least one head, and no head.

Construct sample spaces for one die and for multiple dice, noting 6^n outcomes. Calculate probabilities for even numbers, multiples of three, and their union or intersection when rolling dice.

Learn shortcut and alternative counting methods to find the probability of sums when three dice are rolled, from 3 to 18. See examples like sum 16 equals 1/36.

Explore the algebra of events by constructing new events through union, intersection, and complement within a sample space, illustrated with dice outcomes and De Morgan's laws.

Discover how to build sample spaces for random experiments, using combinations to choose two from five children, and explore outcomes from coin tosses and draws.

Explore mutually exclusive, elementary, and compound events in probability through coin toss and dice examples, learning intersections, null sets, and how sample spaces define outcomes.

Explore probability problems using permutations and arrangements, including vowels together, repeated letters, and round-table neighbor scenarios.

Explore probability problems with balls using combinations and without replacement, including computing at least one green, all green, and two red and two green scenarios.

Apply combinations to probability problems with balls: three drawn from twelve (3 red, 5 green, 4 black) and two-ball draws without replacement, plus distribution of twelve balls into three boxes.

Explore the addition theorem for two events, deriving P(A∪B)=P(A)+P(B)−P(A∩B) from sample space counts, and extend to three events, including mutually exclusive cases.

Apply the addition rule for unions, mutual exclusivity, and complements to compute probabilities like A or B, including intersections and De Morgan concepts, with practical practice problems.

Practice problems on P(A ∪ B) – Part 2 explore union and intersection probabilities through examples like rusted bolts and nuts, multiples, odds, and dice sums, with De Morgan's laws.

Study the addition rule and union of two or more events through practice problems on P(A U B) from a 52-card deck, including two-card draws, red cards, kings, and probability.

Explore conditional probability, including definitions of P(A|B) and P(B|A), and solve problems with dice and two-child scenarios.

Apply conditional probability to two examples: compute P(studies in class 12th | girl) using 430 girls, 43 in 12th, yielding 1/10, and compute P(both odd | sum even) as 3/5.

Practice problems on conditional probability, using dice sums and a fixed first throw, and a test-bank example to compute p(a|b), p(a∩b), and p(a∪b) for easy vs multiple-choice questions.

Explore practice problems on conditional probability that apply P(A|B) using intersection, union, to real cases—red and defective bulbs, two-girl committees, and Hindi/English newspaper readers.

Practice problems illustrate using the multiplication theorem to compute probabilities without replacement, including drawing two even numbers, three good oranges, two kings and an ace, and a white-black-red sequence.

Define independent events as those whose occurrence does not affect another's probability, illustrated by drawing with replacement. Show that P(A ∩ B) = P(A)P(B) and that complements are also independent.

Learn to prove event independence by testing P(E∩F) against P(E)P(F) with card-drawing examples, including spades vs aces, and explore mutual exclusivity and p values.

Learn to solve independent-event problems with replacement, using a box with 10 black and 8 red balls to compute red-red, black-red, and one-of-each probabilities. Use complements and case analysis.

Apply addition and multiplication theorems to probability problems, with independent and dependent events, using two bags of white and black balls, and red queen and black king scenarios.

practice questions apply addition and multiplication theorems to probability, using combinations for drawing cards without replacement and two-bag draws, with results such as 1/50 and 7/15.

Explain odds in favor as probabilities, compute the majority of three independent critics in favor, and apply addition and multiplication rules to determine joint outcomes.

Explore practice problems on addition and multiplication theorems, using conditional probability for drawing without replacement to find the chance the third ball is red, and compute a three-game win scenario.

Practice questions on addition and multiplication theorems in probability, using dice scenarios to derive a and b winning probabilities via geometric-series sums (6/11 vs 5/11; 30/61 vs 31/61).

Apply the law of total probability by partitioning the sample space into regions e1, e2, e3 and summing the conditional probabilities of the event A given each region.

Practice problems on the law of total probability, including two-bag selection by die, two-ball draws to get one red and one black, and gender-based first-class and machine-based defective-bolt probabilities.

Apply the law of total probability to two practice problems: the second ball being blue after the first draw, and white from bag B after transferring from bag A.

Use the law of total probability for transferring balls between bags and drawing from the second bag to determine white-ball probability. Compute results such as 59/130 and 673/1260.

Apply Bayes theorem to revise prior probabilities with conditional likelihoods, using a medical test example to compute the posterior probability that a person has HIV.

Practice problems on Bayes theorem apply conditional probabilities to determine the likelihood that a defective bolt comes from machine B, and that a red ball came from the first bag.

Practice Bayes theorem with urn problems and an accident probability example, computing conditional probabilities when observing white and red draws from bags and identifying the most likely source.

Apply Bayes' theorem and conditional probability to practice problems, including a deck of cards with a lost diamond and a die with truth-telling probabilities, yielding 11/50 and 3/8.

Tackle GMAT-style permutations, combinations, and probability problems to build fundamentals, sharpen analytical thinking, and speed for exam success.

Explore solutions to factorial divisibility, word formation under letter constraints, consecutive green-ball configurations, and four-digit numbers with distinct digits, illustrating permutations, combinations, and basic probability.

Explore counting methods for five-digit numbers from digits 0–5 divisible by three, with and without zero. Apply permutations and combinations to problems on abacus vowels, committees, and medal winners.

Apply combinations and probability to solve committee, photo selection, and shirt-draw problems, highlighting at least one woman and at least one blue shirt, with two throws at a circle-on-square target.

Solve q14 and q15 by using area ratios to find the probability a point in a circle lies inside the inscribed equilateral triangle, and 1/25 for the second yellow ball.

Solve three probability problems using the union formula to find cracked and extra large eggs, at least one even in two draws, and sums equal to 11 from specified sets.