Factorial: Formulas, Properties, Tricks and Examples
Factorial is the product of all positive integers from 1 up to a given non-negative integer. It is written using the symbol !. This page covers factorial formulas, recursive rules, special values, simplification methods, trailing-zero calculations and common factorial questions with numerical examples.
What Is Factorial?
The factorial of zero is defined as 0! = 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorial is defined directly for non-negative integers in standard quantitative aptitude problems.
Factorial Formula & Tricks
Important Formulas
This gives the product of every positive integer from n down to 1, where n is a non-negative integer.
A factorial can be calculated by multiplying n by the factorial of the preceding integer.
This value maintains the recursive rule because 1! = 1 × 0! = 1.
The common factorial terms cancel, leaving r consecutive factors.
Continue the sum while the powers of 5 are less than or equal to n. The result counts factors of 10 because factors of 2 are more abundant than factors of 5 in n!.
Quick Tricks
When factorials appear in a fraction, expand only the unmatched factors instead of calculating the complete factorials.
Count the factors of 5 in n!, including repeated factors such as 25 = 5² and 125 = 5³.
Replace a larger factorial with a smaller factorial multiplied by the intervening integer.
Factorial Concepts
Evaluating Factorials
For example, 4! = 4 × 3 × 2 × 1 = 24. Similarly, 1! = 1 and 0! = 1. Factorial values grow rapidly as n increases.
Recursive and Consecutive Factorial Relations
This relation allows expressions to be combined without fully evaluating the factorial. For example, 5! + 4! = 5 × 4! + 4! = 6 × 4! = 144.
Simplifying Factorial Fractions
If the numerator has the larger factorial, expand it only until the denominator factorial appears. For n ≥ r, n!/(n − r)! contains r factors: n, n − 1, ..., n − r + 1.
Factorial in Permutations and Combinations
nPr counts ordered arrangements of r objects selected from n objects. nCr counts selections where order does not matter. Also, nCr = nC(n − r).
Trailing Zeros in a Factorial
Use ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + ... . Terms are added until the denominator exceeds n. For 50!, the count is ⌊50/5⌋ + ⌊50/25⌋ = 10 + 2 = 12.
Factorial Video Lessons
Watch short topic-wise lessons for quick revision.
Factorials: Trailing Zeros and Prime Powers
Learn how to find trailing zeros in factorials and determine the highest power of a prime that divides a factorial using standard number system methods.
Practice Factorial Questions
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Factorial Quick Quiz
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Factorial Revision Points
Recall these definitions, formulas and calculation rules for factorial questions.
- n! = n × (n − 1) × ... × 2 × 1 for n ≥ 1.
- 0! = 1 and 1! = 1.
- Use n! = n × (n − 1)! to relate consecutive factorials.
- Cancel common factorial terms before multiplying a factorial fraction.
- n!/(n − r)! = n × (n − 1) × ... × (n − r + 1).
- nPr = n!/(n − r)! and nCr = n!/[r!(n − r)!].
- The number of trailing zeros in n! is the sum of the integer parts of n/5, n/25, n/125 and so on.
Factorial FAQs
What is the value of 0!?
0! = 1. This definition makes the recurrence 1! = 1 × 0! valid.
How do you calculate 7! without a calculator?
Multiply the integers from 7 to 1: 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040.
How can 9!/6! be simplified?
Cancel 6!: 9!/6! = (9 × 8 × 7 × 6!)/6! = 9 × 8 × 7 = 504.
What is the formula for the number of trailing zeros in 100!?
Use ⌊100/5⌋ + ⌊100/25⌋ + ⌊100/125⌋ = 20 + 4 + 0 = 24. Therefore, 100! has 24 trailing zeros.
What is the difference between nPr and nCr factorial formulas?
nPr = n!/(n − r)! counts ordered arrangements, whereas nCr = n!/[r!(n − r)!] counts selections where order does not matter.
Is 5! + 4! equal to 9!?
No. 5! + 4! = 120 + 24 = 144, while 9! = 362,880.