# **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: * **Factorial Notation:** The factorial of a non-negative integer *n*, denoted by $n!$, is the product of all positive integers less than or equal to *n*. For example, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$. By definition, $0! = 1$. * **Properties of Factorials:** * $n! = n \times (n-1)!$ * Factorials are fundamental in permutations and combinations for counting the number of ways to arrange or select objects. * **Simplifying Factorial Expressions:** Techniques for canceling out common terms in fractions that involve factorials. * **Solving Factorial Equations:** Methods to find the value of an unknown variable within an equation containing factorials. * **Proofs involving Factorials:** Demonstrating the validity of mathematical statements and identities involving factorials. This often involves algebraic manipulation and applying factorial properties. * **Divisibility Properties of Factorials:** A key property is that $n!$ is divisible by all positive integers up to *n*. --- # **Important Formulas** Below are the key formulas used in this exercise: | Formula | Description | | :------------------------------------------------------- | :------------------------------------------------------------------------------------ | | $n! = n \times (n-1) \times (n-2) \times \dots \times 1$ | Definition of a factorial. | | $n! = n \times (n-1)!$ | Recursive formula for factorials. | | $(2n)! = 2^n n! (1 \cdot 3 \cdot 5 \dots (2n-1))$ | Relationship between the factorial of an even number and the product of odd integers. | | $^n P_r = \frac{n!}{(n-r)!}$ | Formula for permutations. | | $^n C_r = \frac{n!}{r!(n-r)!}$ | Formula for combinations. | | $^n C_r + ^n C_{r-1} = ^{n+1} C_r$ | Pascal's Identity, seen in Question 6(iii). | --- # **Summary** This exercise provides a comprehensive workout in understanding and manipulating factorial expressions. The questions progress from basic evaluation and conversion to more complex proofs and equation solving. A solid grasp of the fundamental definition of a factorial ($n!$) and its recursive property is essential for success. The problems emphasize algebraic simplification, pattern recognition, and the application of factorial identities. Mastering these concepts is crucial for further studies in combinatorics, probability, and advanced mathematics. --- # **Reference Videos**