Become a Permutations and Combinations Master

Master the fundamental principle of counting to combine independent and mutually exclusive choices, using multiplication to compute total outcomes in permutations and combinations contexts.

Define factorial as the product of the first n natural numbers, illustrated by 5! = 5×4×3×2×1 and 8! = 8×7×6×5×4×3×2×1. Explore basic properties of factorials and their computation.

Define permutation as arrangements of some or all items, and apply nPr equals n factorial over (n minus r) factorial with factorial properties, solving a competition-style example to illustrate methods.

Demonstrate the permutation identity n p r = (n-1) p r + r (n-1) p (r-1) by applying result one and factorial rules, showing rhs equals lhs step by step.

Explore permutations through a solved example, applying plus and minus operations and key properties to simplify counting and arrangement.

Explore permutations through solving example 4 and 5, counting signals formed by selecting five from eight flags of different colors.

Master permutations through solved examples 1 and 2 that count distinct words formed from letters with repeats, using factorials and arrangements, illustrated with the word California.

Learn how to count arrangements using permutations through solved examples, such as seating four people at a table and assigning masters, illustrating the fundamental counting principle.

Explore permutations and combinations by solving two worked examples on forming distinct word arrangements from the letters of a word like triangle, with constraints on starting letters.

Explore permutations through a solved example, analyzing position constraints and selection patterns, including specific arrangements like 1 3 5 7 8 in a practical permutation problem.

Explore permutations with five letters by analyzing placements and when letters must stay together, and apply the fundamental principle of counting to determine the number of possible words.

Learn circular permutations through a solved example of seating five boys and five girls around a circle, showing how rotations yield the arrangement and fixing a position yields distinct orders.

Explore circular permutations by solving a seating problem, counting arrangements where a specific person sits on either side of another, and applying the multiplication principle to determine totals.

Master circular permutations by exploring necklace arrangements, rotation and reflection equivalences, and solved examples on beads and flowers in circular order.

Covers circular permutations by illustrating how to count arrangements around a circular table, examining when clockwise and anticlockwise orders are considered different, with solved examples.

Define combinations as selections where order does not matter, compute using nCr and factorials, and explore key results, symmetry, and binomial theorem connections.

Explore combinations with solved example 1 and 2 from the permutations and combinations master course, using given values and notations to derive results.

Master permutations and combinations through solved examples 3 and 4, as presented in this lecture, and apply core techniques to solving combination problems.

Become a permutations and combinations master by studying solved example 5 and 6 of combinations, including selecting 15 from 50.

Explore counting combinations with constraints using C(n, k) notation, through solved examples like selecting 11 players from 15 with one excluded and choosing six from 11 with two excluded.

Master combinations through solved examples, applying the fundamental principle of multiplication to count one-or-more selections, such as inviting friends or choosing items.

Apply rule three of combinations to an at-most-n books problem from two n plus one books, deduce 63 equals the binomial sum and find n equals 3.

Explore combinations with identical items and learn how to count possible selections, illustrated by solved examples of choosing items from a set.

Explore combinations through solved examples and apply the fundamental principles of counting to determine how many ways one or more items can be selected from a set.

Apply number theory concepts to a solved example, exploring divisors, prime factorization, coprime numbers, and counting formulas to determine divisor-related counts.

work through solved example 2 on applying permutations and combinations to number theory, setting up the problem, manipulating expressions, and calculating counts step by step.

Explore an application of permutations and combinations to number theory by counting ways to split a number into two coprime parts, using gcd and related formulas.

Explore dividing objects into two or three groups using permutations and combinations, and apply the multiplication principle to count divisions for equal and varying group sizes.

Learn how to divide a set of 52 into four equal groups using the division formula, with solved example 1 and 2.

Explore division into groups with a solved example using a 52-card deck and selecting 17 cards. Use the 52 choose 17 combinations formula, noting order does not matter.

Divide twelve items into groups of five and seven using the division into groups method, noting when order matters versus not, and extend to groups of five, four, and three.

Work through arrangement in groups case 1, solved example 1, distributing five distinct balls into three distinct boxes with no empty boxes using the second formula for combinations.

Count arrangements of five distinct balls into three distinct boxes with at least one ball per box, using case 2 of arrangement in groups from the permutations and combinations course.

Investigate counting distributions of identical items into distinct groups using combinations, with and without empty groups, through solved examples in arrangement in groups.

Explore the arrangement in groups case 3 by analyzing when x and y are nonnegative, applying inclusion rules for groups, and counting solutions.

Explore arrangement in groups through case 3, solved examples 4 and 5, and count solutions using a system of equations and the fundamental principle of multiplication.

Analyze solved example 6 in arrangement in groups, case 3, to sharpen your skills in permutation and combination reasoning and solving equations with variables x, y, z.

Explore arrangement in groups through solved examples 1 and 2, applying permutations, combinations, and the binomial theorem to count groupings and expand expressions.

Study arrangement in groups through solved example 1 from case 5, using binomial coefficients and the second form to determine the group configurations.

Explore the derangements concept with a solved example using inclusion-exclusion to count permutations with no fixed points, including cases with at least two wrong letters.

Explore the multinomial theorem and its expansion to enumerate combinations and coefficients, illustrated by a word examination example selecting letters from examination.

Explore the multinomial theorem and how to count the number of solutions of an equation through explicit expansions and a solved example.

Explore how the multinomial theorem counts solutions to equations, using a solved example to illustrate counting with variables not equal.