Permutations and combinations Tutors

Permutations and combinations form a branch of combinatorics in Mathematics. Permutations count ordered arrangements (nPr: n Permutation r), combinations count unordered selections (nCr: n Choose r). For example, arranging 3 books on a shelf uses permutations, choosing 2 toppings on a pizza uses combinations, using factorial formulas.

Popular alternative names include - Permutations: arrangements, ordered samples, rearrangements - Combinations: selections, unordered samples, subsets

Major topics/subjects in permutations and combinations - Fundamental principle of counting (product rule, sum rule) - Factorials and factorial notation - Permutations of distinct objects and with repetition - Circular permutations (seating arrangements around a table) - Combinations of distinct objects and with repetition (“stars and bars” method) - Multiset permutations (e.g., arranging letters in MISSISSIPPI) - Binomial coefficients and Pascal’s triangle - Inclusion‑exclusion principle (such as counting derangements) - Applications in probability and statistics (e.g., lottery odds)

A brief history of most important events in permutations and combinations Early combinatorial ideas appear in 6th century BCE Indian works by Pingala on poetic meters. Chinese mathematician Jia Xian (ca. 11th century) developed versions of Pascal’s triangle. In 1653 Blaise Pascal formalized binomial coefficients; around the same time Fermat and Pascal laid foundations of probability via combinatorial arguments. Christian Huygens (1657) studied arrangements, while Jacob Bernoulli (1685) applied them to games of chance. Abraham de Moivre’s 1718 “The Doctrine of Chances” linked combinatorics to probability distributions. By the 19th century, August De Morgan and George Boole expanded counting methods, and today they underpin algorithms, cryptography and coding theory—even everyday UI designs for seating or scheduling involve these ideas.

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Permutations and combinations stand out because they let us count ways to arrange or select items without having to list every possibility. They focus on order and grouping, giving clear formulas to solve many real-world problems in probability, scheduling or design. This unique power to predict outcomes and model scenarios makes them a key tool in mathematics and exams.

Compared to other math topics, permutations and combinations are easy to use once you learn the basic formula rules, so they build strong logical thinking and help in many subjects like statistics or computer science. On the downside, students often find their notation and conditions confusing at first, and mastering all cases can feel time-consuming until the patterns become familiar.

After mastering permutations and combinations, students can study advanced combinatorics, discrete mathematics, probability theory or graph theory. They can also pursue degrees in computer science, data science and artificial intelligence. Graduate programs in operations research and cryptography rely heavily on counting methods to solve complex problems.

Popular roles include data analyst, algorithm engineer, quantitative analyst at financial firms, actuary in insurance, cryptographer in cybersecurity, and operations research consultant. In these positions, professionals design counting models, optimize resource allocation, analyze risks and build efficient algorithms or security systems.

We study permutations and combinations to develop strong problem‑solving and logical reasoning skills. Test preparation for exams like SAT, GRE, math olympiads or engineering entrance tests often covers these topics. Mastery helps answer questions on probability and arrangement quickly and accurately.

Applications extend to network design, task scheduling, resource allocation, genetic sequencing, coding theory, and ticketing systems. These methods help count possibilities, calculate probabilities and support decision making in logistics, finance, computer science, biology and cybersecurity. They also form the basis of combinatorial optimization.

Start by learning the basic ideas: understand what factorial (!) means, then nPr (ordered choices) and nCr (unordered choices) formulas. Break it into steps—memorize formulas, work simple examples (like arranging 3 books or choosing teams), then try tougher ones (with repetition or restrictions). Use flashcards for formulas, set small goals (10 problems a day), and review mistakes to build confidence.

Permutations and combinations can seem tricky at first because of formulas and casework, but they follow clear rules. If you practice regularly and understand why each formula works, you’ll find patterns and logic that make problems easier over time.

You don’t have to hire a tutor if you’re self‑motivated and use good resources—videos, textbooks and problem sets can guide you. But if you hit roadblocks, a tutor speeds up your learning by spotting errors, offering tips and keeping you on track.

MEB offers 24/7 online one‑to‑one tutoring, homework and assignment help in permutations and combinations. Our expert tutors tailor lessons to your pace, clear doubts on the spot and give practice problems until you’re confident—all at affordable rates.

Most students spend about 1–2 weeks mastering the core formulas and simple problems, plus another 2–4 weeks on advanced questions and tricky cases. Consistent daily practice (30–60 minutes) usually leads to solid understanding in about a month.

YouTube: Khan Academy (Permutations & Combinations), PatrickJMT, Math Antics. Websites: Brilliant.org, MathIsFun.com, PurpleMath.com, Coursera combinatorics courses. Books: Schaum’s Outline of Combinatorics (Saraswat), A First Course in Probability (Ross), Concrete Mathematics (Graham, Knuth & Patashnik), Permutations & Combinations (Agarwal), Introduction to Probability (Grimmett & Stirzaker).

College students, parents and tutors in the USA, Canada, UK, Gulf and beyond—if you need a helping hand, whether it’s 24/7 online tutoring or assignment support, our MEB tutors are here to help at an affordable fee.