Master the concept of Factorial of a Number for CBSE Class 11 Applied Mathematics. Learn definitions, properties, and solve practice problems.
The factorial of a non-negative integer n, denoted by n! or |n, is the product of all positive integers less than or equal to n. For example, 4! = 4 × 3 × 2 × 1 = 24. By definition, the factorial of zero is defined as 0! = 1.
Step 1: Write the product: 5! = 5 × 4 × 3 × 2 × 1.
Step 2: Multiply the numbers: 5 × 4 = 20; 20 × 3 = 60; 60 × 2 = 120; 120 × 1 = 120.
Answer: 120
Factorials exhibit a recursive structure, meaning any factorial can be expressed in terms of a smaller factorial. This property is essential for simplifying complex algebraic expressions involving factorials. It allows us to cancel common terms in fractions.
Step 1: Expand 7! as 7 × 6 × 5!.
Step 2: Rewrite the expression: (7 × 6 × 5!) / 5!.
Step 3: Cancel 5! from numerator and denominator: 7 × 6 = 42.
Answer: 42
When performing arithmetic operations with factorials, one must evaluate each factorial separately before adding or subtracting. Note that (a + b)! is generally not equal to a! + b!. Always simplify expressions using the recursive property first.
Step 1: Calculate 4! = 4 × 3 × 2 × 1 = 24.
Step 2: Calculate 3! = 3 × 2 × 1 = 6.
Step 3: Add the results: 24 + 6 = 30.
Answer: 30