Factorial Calculator | What is the factorial of 100?

What is Factorial?

The factorial of a number is the product (multiplication) of all positive whole numbers less than or equal to that number. It’s like counting down, starting from the number itself and multiplying by each smaller number until you reach 1. This is symbolized with an exclamation mark !.

Formula for Factorial:

• n!=n×(n−1)×(n−2)×⋯×2×1n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1n!=n×(n−1)×(n−2)×⋯×2×1

For example:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

Let’s Break it Down with Some Examples:

Example 1: 3!

• 3!=3×2×1=63! = 3 \times 2 \times 1 = 63!=3×2×1=6
• So, the factorial of 3 is 6.

Example 2: 4!

• 4!=4×3×2×1=244! = 4 \times 3 \times 2 \times 1 = 244!=4×3×2×1=24
• For 4, the factorial is 24.

Example 3: 5!

• 5!=5×4×3×2×1=1205! = 5 \times 4 \times 3 \times 2 \times 1 = 1205!=5×4×3×2×1=120

What About Zero? What is 0!?

• 0! = 1
• This might seem strange, but this is a special rule in math! The reason behind this is related to combinations and permutations, which we'll get to in a moment.

Why is 0! = 1?

This might seem confusing, but here’s the key idea:

• In mathematics, factorials are often used in combinations (grouping things) and permutations (arranging things).
• When you have 0 objects to arrange or group, there's exactly 1 way to do it: by doing nothing. This is why 0! is defined as 1.

How to Find Factorial for Larger Numbers:

The key to finding the factorial of larger numbers is to multiply the number by the factorial of the number before it. Let’s take an example.

Example 4: Finding 6!

You know from earlier:

• 5!=5×4×3×2×1=1205! = 5 \times 4 \times 3 \times 2 \times 1 = 1205!=5×4×3×2×1=120

Now, to find 6!:

• 6!=6×5!=6×120=7206! = 6 \times 5! = 6 \times 120 = 7206!=6×5!=6×120=720

So, 6! is 720.

Factorials and Their Applications in Math:

Factorials are super important in several areas of mathematics, especially in combinatorics (how things can be arranged or grouped).

Permutations:

A permutation is an arrangement of things in a particular order. The formula to calculate a permutation is:

• nPr = n! / (n - r)!

Where:

• n is the total number of items.
• r is the number of items you're choosing.

Example: You have 10 books, and you want to know how many ways you can arrange 3 of them on a shelf. You use the permutation formula.

• 10P3 = 10! / (10 - 3)! = 10! / 7!
• You can simplify this:
  10P3=10×9×8×7!7!10P3 = \frac{10 \times 9 \times 8 \times 7!}{7!}10P3=7!10×9×8×7!​
  So, 10P3 = 10 × 9 × 8 = 720.

Thus, there are 720 ways to arrange 3 books out of 10.

Combinations:

A combination is a way of selecting items without considering the order. The formula to calculate combinations is:

• nCr = n! / [r!(n - r)!]

Where:

• n is the total number of items.
• r is the number of items you are choosing.

Example: You want to choose 3 books from a set of 10, but the order of the books doesn’t matter. You use the combination formula.

• 10C3 = 10! / [3!(10 - 3)!] = 10! / (3! × 7!)
• Simplify:
  10C3=10×9×8×7!(3×2×1)×7!10C3 = \frac{10 \times 9 \times 8 \times 7!}{(3 \times 2 \times 1) \times 7!}10C3=(3×2×1)×7!10×9×8×7!​
  So, 10C3 = 120.

Thus, there are 120 ways to choose 3 books from 10 without caring about the order.

Factorial Growth – The Size of Factorials:

As you increase the number, factorials grow really fast!

• 5! = 120
• 10! = 3,628,800
• 20! = 2,432,902,008,176,640,000

For numbers as large as 100, 100! has 158 digits and is such a large number that it’s hard to even write out. It’s approximately:
93,326,215,443,944,152,681,699,238,856,266,700,490,715,968,264,381,621,468,592,963,895,217,599,993,229,915,608,941,463,976,156,518,286,253,697,920,827,223,758,251,185,210,916,864,000,000,000,000,000,000,000,000.

Negative Factorials:

Factorials don’t work for negative numbers. You can’t find the factorial of -1, -2, or any negative number. Why? Because multiplying by negative numbers leads to undefined results. Factorials are only for non-negative whole numbers (0, 1, 2, 3, 4…).

Important Facts to Remember:

1. n! = n × (n - 1) × (n - 2) × ... × 1
2. 0! = 1 (This is a special rule in math)
3. Factorials grow really fast as the number gets bigger.
4. You can use factorials to solve permutations and combinations, which are ways to count different arrangements and selections of things.
5. Factorials are important in fields like probability, statistics, and discrete mathematics.

Fun Fact:

In history, Christian Kramp, a French mathematician, introduced the symbol for factorial n! in 1808. It was a big step in making math easier to write and understand.