What Is the Factorial of 100, and How Is It Calculated?

The factorial of 100, written 100!, is the product of every whole number from 1 through 100. It is a 158-digit integer, approximately 9.332621544394415 × 10¹⁵⁷. The exclamation mark is mathematical notation, not punctuation added for emphasis.

Here is the exact value. It is shown without commas so you can copy it into a tool that accepts large integers:

93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000

That long result answers the calculation, but understanding the operation is more useful than memorizing its digits. Factorials explain how quickly the number of possible arrangements grows, why some counting problems need division, and why a calculator may display only an approximation.

What does the factorial symbol mean?

For a positive whole number, factorial means multiplying downward to 1. Thus 5! = 5 × 4 × 3 × 2 × 1 = 120. For 100!, the same rule becomes 100 × 99 × 98 × … × 3 × 2 × 1. The dots stand for the intervening whole-number factors, not a shortcut that omits them.

Multiplication order does not change the result. Starting at 1 and working upward gives the same integer as starting at 100 and working downward. Wolfram MathWorld’s factorial reference gives the definition and the special value 0! = 1.

Do not confuse 100! with 100 × 100 or 100¹⁰⁰. The square is 10,000. The power multiplies 100 by itself 100 times, producing 10²⁰⁰. Factorial instead uses a different factor at each step, so its value is much smaller than that power, while still enormous.

How can you calculate the exact value?

A direct method keeps a running product. Start with 1, multiply by 2, then by 3, continuing until 100. At the fourth step the result is 24; at the fifth it is 120. The procedure is simple even though the number soon becomes too long to handle comfortably by hand.

This short Python 3 example performs that calculation using integers:

result = 1
for number in range(1, 101):
    result *= number

print(result)
print(len(str(result)))  # 158 digits

The upper limit in range(1, 101) is excluded, so the final multiplier is 100. Starting result at zero would make every later product zero. Starting at one preserves the product, and including the factor 1 makes the intended range easy to see.

Python’s integer type supports unlimited precision, subject to available memory. This lets the calculation retain every digit. Keep the values as integers; converting the result to a floating-point number changes the representation and can lose low-order digits.

Why does a calculator show an E or a power of ten?

A display such as 9.332621544E157 uses scientific notation. The E means “times ten to the power of,” so E157 represents × 10¹⁵⁷. It does not mean an error. The shown leading digits are rounded to fit the display; they are not the entire exact answer.

Because the result is between 10¹⁵⁷ and 10¹⁵⁸, it has 158 decimal digits. The exponent is one less than the digit count for a positive integer written in normalized scientific notation. If you need only the scale, the rounded form is convenient. If you need exact digits or integer arithmetic, use the full integer.

Why does 100! end with 24 zeros?

Each trailing zero comes from a factor of 10, and 10 is 2 × 5. In the product from 1 to 100 there are more factors of 2 than factors of 5, so count the fives. The multiples 5, 10, 15, and so on through 100 supply 20 of them.

Four of those multiples, 25, 50, 75 and 100, each contain a second factor of 5. That adds four more. There are no multiples of 125 in the range, so no third layer is needed. The total is 20 + 4 = 24 trailing zeros, matching the exact result above.

Counting only the multiples of 10 would miss zeros formed by factors coming from different numbers. For example, multiplying 2 by 5 creates a factor of 10 even though neither starting number ends in zero. This is why counting prime factors is more reliable than inspecting the original numbers’ endings.

What does 100! count in a real problem?

Suppose you have 100 distinct numbered cards and want to arrange all of them in one row. You have 100 choices for the first position, 99 for the next, and one fewer at each later position. Multiplying those choices gives 100! possible orders. This is the basic permutation interpretation.

The word distinct matters. Swapping two identical unmarked objects may not create a new visible arrangement. Likewise, arranging only some objects, arranging around a circle, or imposing position restrictions changes the counting problem. Decide what counts as a different outcome before selecting a formula.

For a smaller hands-on example, arrange four differently colored objects in a row. Each choice of first object leaves three choices for second place, two for third and one for last: 4 × 3 × 2 × 1 = 24 arrangements. Listing those 24 is manageable; listing all arrangements of 100 objects is a very different task.

When should you cancel factorials instead of expanding them?

If a problem asks for 100! divided by 98!, do not calculate both huge numbers first. Write 100! as 100 × 99 × 98!. The shared 98! cancels, leaving 100 × 99 = 9,900. This counts ordered selections of two different objects from 100.

If order does not matter, each pair has been counted twice: choosing card A then B gives the same pair as choosing B then A. Divide by 2 to get 4,950 pairs. This is “100 choose 2,” a binomial coefficient, rather than 100!.

How can you check your answer?

Check the operation first, then the result’s size and ending. For 100!, expect 158 digits and exactly 24 trailing zeros. These checks can catch a transcription error, but they cannot prove every middle digit is correct. For exact work, compare the integer produced by two independent calculation methods.

Keep the distinction between an exact value and a rounded display clear when sharing your answer. Write an approximation sign or say “approximately” for scientific notation with shortened digits. For practice with smaller factorials and counting exercises, you can browse combinatorics workbooks on Amazon; a pencil and paper are enough to begin.