Civil Service Exam (Professional) Numerical Ability — Permutation & CombinationStudy Notes

Before diving into permutations and combinations, we must first understand factorials. A factorial, denoted by the symbol '!', represents the product of all positive integers from 1 up to and including that number. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are the building blocks of counting techniques and appear in both permutation and combination formulas. By convention, 0! = 1, which is important to remember when solving problems. Factorials grow very rapidly - while 3! = 6, we have 10! = 3,628,800. This rapid growth helps explain why there are so many possible arrangements and selections in counting problems.

Permutation is a mathematical technique used to count the number of ways to arrange objects where the order of arrangement is important. Think of it as organizing a line of students for a parade - changing the position of any student creates a completely different arrangement. The formula for permutation when selecting r objects from n total objects is nPr = n!/(n-r)!. This formula ensures we account for all possible arrangements while avoiding impossible selections (when r > n). Permutations are commonly used in situations involving rankings, seating arrangements, password creation, and any scenario where position or sequence matters.

Combination is a counting technique used when we want to select objects from a group, but the order of selection is not important. Unlike permutation, combinations treat different arrangements of the same objects as identical. For example, selecting team members for a basketball team - whether you choose John first or Mary first doesn't matter; what matters is who is on the team. The combination formula is nCr = n!/(r!(n-r)!). Notice that this formula includes an additional r! in the denominator compared to permutation, which accounts for eliminating the different orders of the same selection.

The fundamental difference between permutation and combination lies in whether order matters. In permutations, ABC and BAC are different arrangements, while in combinations, they represent the same selection. To decide which technique to use, ask yourself: 'Does changing the order create a different outcome?' If yes, use permutation; if no, use combination. Real-world applications help clarify this distinction: arranging books on a shelf requires permutation (order matters), while selecting books to read requires combination (order doesn't matter). Race positions use permutation (1st, 2nd, 3rd are different), while team selection uses combination (team members are equal).

Successful problem-solving in permutations and combinations requires careful reading and identification of key terms. Look for words like 'arrange,' 'order,' 'rank,' or 'sequence' to indicate permutation, and words like 'choose,' 'select,' 'pick,' or 'group' to suggest combination. Common mistakes include confusing when to use each technique, making calculation errors with factorials, and misinterpreting problem constraints. Always check if repetition is allowed, as this changes the approach entirely. Practice with various problem types helps build intuition for recognizing patterns and choosing the correct method.

Permutations and combinations appear frequently in Philippine competitive exams like UPCAT, CSE, and professional licensure tests. These concepts apply to probability calculations, statistical analysis, computer science algorithms, and business decision-making. In probability, combinations help calculate the number of favorable outcomes, while permutations determine total possible arrangements. Understanding these concepts also helps in advanced mathematics topics like binomial expansion and discrete mathematics. Regular practice with exam-style problems builds confidence and accuracy in identifying and solving counting problems efficiently.