Exercise 7.1 – Factorial (First Year Math – New Syllabus 2025)

 Exercise 7.1 – Factorial (First Year Math – New Syllabus 2025)

Introduction:

In mathematics, the symbol n! (read as "n factorial") is very important. At first, the exclamation mark (!) looks like something we use in language, but in mathematics it has a special meaning.

The factorial is widely used in algebra, probability, combinatorics, binomial theorem, calculus, and even higher mathematics. It helps us to calculate the number of arrangements, the number of ways to choose objects, and it also appears in many mathematical series and formulas.

In Exercise 7.1 of First Year Mathematics (New 2025 Edition), we study factorial in detail. The main goals of this exercise are:

1. To understand the definition of factorial.

2. To learn how to expand factorial expressions.

3. To prove that 0! = 1.

Let us go step by step.

Definition of Factorial

For any positive integer n, the factorial of n (written as n!) is the product of all positive integers from 1 up to n.

Formula:

n! = n × (n - 1) × (n - 2) × ... × 3 × 2 × 1

Examples:

1! = 1

2! = 2 × 1 = 2

3! = 3 × 2 × 1 = 6

4! = 4 × 3 × 2 × 1 = 24

5! = 5 × 4 × 3 × 2 × 1 = 120

So factorial grows very quickly. For example:

10! = 3,628,800

20! = 2,432,902,008,176,640,000

Recursive Definition of Factorial

Factorial can also be defined recursively.

n! = n × (n - 1)!   for n ≥ 1

with the condition that:

0! = 1

Example using recursion:

4! = 4 × 3!

3! = 3 × 2!

2! = 2 × 1!

1! = 1 × 0!

This shows that for factorials to work correctly, we must accept that 0! = 1.

Why 0! = 1 ?

This is a very common question among students. If factorial means "multiply all numbers from n down to 1", then why is 0! not zero? Why is it equal to 1?

The answer comes from logic, formulas, and applications.

(1) Factorial and Arrangements (Permutations)

The number of ways to arrange n objects = n!

For 1 object, the number of arrangements = 1! = 1

For 0 objects, there is exactly one way to arrange them – by doing nothing.

So, 0! must be equal to 1.

(2) Consistency with Recursive Formula

From recursive definition:

n! = n × (n - 1)!

If n = 1:

1! = 1 × 0!

We know that 1! = 1. So:

1 = 1 × 0!

This gives:

0! = 1

So, for the recursive formula to remain correct, we must define 0! = 1.

(3) Combinations Formula

The number of ways to choose r objects from n objects is:

Cr = n! / ( r! × (n - r)! )

If n = r, then:

nCn = n! / ( n! × 0! ) = 1 / 0!

But we already know nCn = 1.

So: 1 / 0! = 1 → 0! = 1

(4) Empty Product Rule

In mathematics, multiplying "no numbers" is called an empty product.

By definition, an empty product = 1 (not 0).

So, since factorial of 0 means multiplying no numbers, we get:

0! = 1

PDF SOLUTION:

Applications of Factorial

Factorials are used in many areas of mathematics:

1. Permutations – number of arrangements of n objects = n!

Example: Arrangements of 5 books = 5! = 120

2. Combinations – formula involves factorials.

Example: 5C2 = 5! / (2! × 3!) = 10

3. Binomial Theorem – expansion coefficients involve factorials.

4. Probability – factorials are used to calculate possible outcomes.

5. Series Expansions – exponential and trigonometric series use factorials.

Example: e^x = 1 + x/1! + x^2/2! + x^3/3! + ...

6. Higher Mathematics – factorials appear in calculus, gamma function, and advanced topics.

Common Mistakes Students Make

1. Thinking factorial means just multiplying by 1 (e.g., 5! = 5 × 1 = 5). This is wrong.

2. Forgetting that 0! = 1 (many think it is 0).

3. Mixing permutations and combinations.

4. Confusing factorial with power (n! ≠ n^n).

Summary

Factorial means the product of natural numbers from 1 up to n.

n! = n × (n - 1) × ... × 3 × 2 × 1

Recursive definition: n! = n × (n - 1)!

For consistency, we define 0! = 1

Factorials are used in permutations, combinations, binomial theorem, probability, and higher mathematics.

#Factorial #Exercise7_1 #FirstYearMath #MathSyllabus2025 #NewMathBook #IntermediateMath #MathLearning #MathPractice #MathExercise #MathProblems #FactorialMath #CollegeMath #MathEducation #MathUniverse #MathForBeginners #StudyMath #LearnFactorial #1stYearMath2025