Breaking News

# What is 40 factorial ?

Steps to calculate factorial of 40

To find 40 factorial, or 40!, simply use the formula that multiplies the number 40 by all positive whole numbers less than it.

Let’s look at how to calculate the Factorial of 40:

40! is exactly :
815915283247897734345611269596115894272000000000
Factorial of 40 can be calculated as:
40! = 40 x 39 x 38 x 37 x 36 x 35 x 34 x 33 x 32 x 31 x 30 x 29 x 28 x 27 x 26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

## What is Factorial?

The concept of factorial in mathematics pertains to a value obtained by multiplying a series of descending natural numbers. It is denoted by an exclamation mark after a number. For example, when we write ‘5!’, it means the product of all positive integers from 5 down to 1. The significance of understanding the factorial of 40, or ’40!’, stems from its vast applications in permutations, combinations, and various probability scenarios. It provides the total number of ways in which a set of 40 different elements can be arranged.

## Formula to Calculate the Factorial of 40

To understand the factorial of any number, we begin with the basic formula which states that n! equals the product of all natural numbers from n down to 1. Hence, for 40 factorial, the process would involve:

40! = 40 × 39 × 38 × … × 3 × 2 × 1

This continues until every number down to 1 has been included in the multiplication. While the value of 40! is far too large to detail each step, this example illustrates the initial sequence that would be followed in its calculation.

## What is the Factorial of 40 Used For?

The concept of factorial is indispensable in disparate fields of mathematical study and practical application. Specifically, the factorial of 40 could be used in:

• Combinatorics, to determine the number of possible combinations in a set of 40 elements.
• Probability theory, for calculating the likelihood of different outcomes in a sample space with 40 elements.
• Quantitative problems in business situations such as resource management or logistics planning involving 40 different options or constraints.

## Exercises

1. How many zeros are at the end of the factorial of 40?
2. If you are arranging 40 books on a shelf, in how many different ways can they be ordered?

## Solutions to Exercises

1. The factorial of 40 will end in the same number of zeros as there are pairs of 2 and 5 in its prime factors. In 40!, there are more multiples of 2 than multiples of 5, so we count the number of times 5 will occur, which is 8 (since 40/5 = 8), indicating that 40! ends with 8 zeros.
2. 40 books can be arranged in 40! different ways. It illustrates the immense number of permutations that can be achieved when ordering a set of 40 distinct items.

### Q: What is the exact value of 40 factorial?

A: The exact value of 40 factorial is a number with 48 digits: 8.15915283247898e+47. Due to its large size, 40! is often approximated or only its properties, such as trailing zeros, are considered.

### Q: Why is the factorial of 0 equal to 1?

A: Zero factorial is defined as 1 to maintain the consistency of the factorial’s property that any number n! is equivalent to (n+1)! divided by (n+1). Therefore, 1! divided by 1 equals 1, and for this relationship to hold, 0! must also be defined as 1.

### Q: Can you have a factorial of a negative number?

A: No, factorials are undefined for negative numbers because they rely on the product of natural numbers down to 1, and this sequence does not include negative integers.