Permutation, Combination and Probability

This chapter builds the foundation of counting techniques and uncertainty, which are heavily tested through both short questions and long descriptive problems in board examinations. The journey begins with the Fundamental Principle of Counting (multiplication and addition rules), which dictates that if an operation can be performed in $m$ ways and a second independent operation in $n$ ways, both can be performed in $m \times n$ ways. Students must master the concept of Factorial Notation ($n!$) and its algebraic manipulations, as simplifying factorial expressions is a staple of short-answer sections.

The study of arrangement leads directly to Permutations, defined as an ordered arrangement of objects. The primary formula, $P(n, r) = \frac{n!}{(n-r)!}$, is applied when order matters. Board examiners frequently test permutations involving restrictions, such as arrangements where certain objects must always come together, or never together, as well as circular permutations where $(n-1)!$ is applied for linear objects arranged in a circle. Special care must be taken with permutations of like objects, using $\frac{n!}{n_1! n_2! \dots n_k!}$, which frequently appears in word-problem formats like arranging letters of words (e.g., "MATHEMATICS" or "PARALLEL").

When the selection order is immaterial, we transition to Combinations, given by $C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$. Students must thoroughly memorize and apply key combinatorial identities, most notably $\binom{n}{r} = \binom{n}{n-r}$ and $\binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r}$, which are popular targets for analytical proof questions in the long-answer section. Board problems often require distinguishing between when to use a permutation versus a combination (e.g., forming committees, selecting cards from a deck, or choosing geometry questions from a paper).

The final segment introduces Probability using the classical definition: $P(E) = \frac{n(E)}{n(S)}$, where $n(E)$ is the number of favorable outcomes and $n(S)$ is the total sample space. Students must be adept at utilizing permutation and combination formulas to calculate $n(E)$ and $n(S)$ in complex scenarios, such as drawing colored balls from an urn. Essential theorems that must be practiced include the Addition Law of Probability for mutually exclusive and non-mutually exclusive events $[P(A \cup B) = P(A) + P(B) - P(A \cap B)]$, Conditional Probability, and the Multiplication Law for Independent Events. Mastering these rules ensures full marks in the probability portion of the board paper.