# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- --- # **Key Concepts** This exercise focuses on the following concepts: * **Combination:** A selection of items from a collection, such that the order of selection does not matter. It is about choosing a subset. * **Combination Formula:** The fundamental formula for calculating the number of ways to choose *r* items from a set of *n* distinct items. * **Properties of Combinations:** Understanding key identities is crucial for solving equations and proofs. * **Symmetry:** ${}^{n}C_r = {}^{n}C_{n-r}$, which simplifies many problems where the number of items to be excluded is smaller than the number to be chosen. * **Pascal's Identity:** ${}^{n}C_r + {}^{n}C_{r-1} = {}^{n+1}C_r$, a recursive relationship fundamental to Pascal's Triangle. * **Solving Combination Equations:** Using algebraic manipulation and the properties of combinations to find unknown values of *n* or *r$. * **Combinations with Constraints:** Applying the combination formula to scenarios with specific conditions, such as: * **"At least" / "At most":** Requiring the calculation of multiple cases or using the complement principle. * **Inclusion/Exclusion:** Problems where a particular item must be included or is excluded. * **Grouping:** Problems involving division of a larger set into smaller subgroups. * **Geometric Applications:** Using combinations to count geometric figures like lines, triangles, and diagonals that can be formed from a set of points. --- # **Important Formulas** Below are the key formulas used in this exercise: | Formula | Description | | :------------------------------------------------- | :------------------------------------------------------------------------------------------------------------------- | | ${}^{n}C_r = \frac{n!}{r!(n-r)!}$ | The number of combinations of *n* distinct objects taken *r* at a time. | | If ${}^{n}C_p = {}^{n}C_q$, then $p=q$ or $p+q=n$. | The symmetry property of combinations. | | ${}^{n}C_r + {}^{n}C_{r-1} = {}^{n+1}C_r$ | Pascal's Identity, a fundamental recursive formula. | | $r \cdot {}^{n}C_r = n \cdot {}^{n-1}C_{r-1}$ | A useful identity for simplifying expressions and proofs. | | Number of diagonals = ${}^{n}C_2 - n$ | Formula for the number of diagonals in an *n*-sided polygon. | | Total Ways = Ways (Case 1) + Ways (Case 2) + ... | The sum rule for mutually exclusive cases (e.g., committees with "at least" or "at most" conditions). | | Required Ways = Total Ways - Unwanted Ways | The complement principle (e.g., finding committees with "at least one man" by subtracting committees with "no men"). | --- # **Summary** This exercise provides a thorough grounding in the theory and application of combinations. It starts with the formal proofs of essential combination identities, which build a strong theoretical foundation. The subsequent questions focus on solving for variables *n* and *r* within combination equations, reinforcing algebraic skills. The core of the exercise lies in applying these concepts to a variety of word problems, including forming committees, selecting teams, solving geometric counting problems, and dividing groups. A key learning is how to translate constraints like "at least," "at most," and other specific conditions into mathematical operations involving combinations. Mastering this topic requires a clear understanding of when to use combinations (order doesn't matter) versus permutations and how to break down complex problems into simpler, calculable parts. --- # **Reference Videos**