# Factorial

# Factorial

The number n! is the product 1×2×3×4×…× n, that is, the product of all the natural numbers from 1 up to n, including n itself where 1 is a natural number. It is called either n factorial or factorial n. Thus 5! is the number 1×2×3×4×5, or 120.

Older books sometimes used the symbol In for n factorial, but the numeral followed by an exclamation point is currently the standard symbol. In 1808, French mathematician Christian Kramp (1760–1826) first used the notation n!.

Factorials show up in many formulas of statistics, probability, combinatorics, calculus, algebra, and elsewhere. For example, the formula for the number of permutations of n things, taken n at a time, is simply n!. If a singer chooses eight songs for his or her concert, these songs can be presented in 8!, or 40,320 different orders. Similarly, the number of combinations of n things r at a time is n! divided by the product r!(n-r)!. Thus, the number of different bridge hands that can be dealt is 52! divided by 13!39!. This happens to be a *very* large number.

When used in conjunction with other operations, as in the formula for combinations, the factorial function takes precedence over addition, subtraction, negation, multiplication, and division unless parentheses are used to indicate otherwise. Thus, in the expression r!(n-r)!, the subtraction is done first because of the parentheses; then r! and (r-n)! are computed; then the results are multiplied.

As n! has been defined, 0! makes no sense. However, in many formulas, such as the one above, 0! can occur. If one uses this formula to compute the number of combinations of 6 things 6 at a time, the formula gives 6! divided by 6!0!. To make formulas like this work, mathematicians have decided to give 0! the value 1. When this is done, one gets 6!/6!, or 1, which is, of course, exactly the number of ways in which one can choose all six things.

As one substitutes increasingly large values for n, the value of n! increases very fast. Ten factorial is more than three million, and 70! is beyond the capacity of even those calculators which can represent numbers in scientific notation.

This is not necessarily a disadvantage. In the series representation of sine x, which is x/1! - x3/3! + x5/5! -…, the denominators get large so fast that very few terms of the series are needed to compute a good decimal approximation for a particular value of sine x.

# Factorial

# Factorial

The pattern of multiplying a positive **integer** by the next lower consecutive integer occurs frequently in mathematics. Look for the pattern in the following expressions.

7 × 6 × 5 × 4 × 3 × 2 × 1

4 × 3 × 2 × 1

(*n* + 5) × (*n* + 4) × (*n* + 3) × (*n* + 2) × (*n* + 1) × *n*

The mathematical symbol for this string of factors is the familiar exclamation point (!). This pattern of multiplied whole numbers is called *n* factorial and is written as *n* ! So, starting with the greatest factor, *n,* the factorial pattern is as follows:

*n* ! = *n* (*n* - 1)(*n* - 2)(*n* - 3)… (1).

So,

3! is 3 × 2 × 1 = 6

5! is 5 × 4 × 3 × 2 × 1 = 120 and 1! = 1.

Zero factorial (0!) is arbitrarily defined to be 1.

Most scientific calculators have a key (such as *x* !) that can be used to find factorial values. As *n* becomes larger, the value of its factorials increases rapidly. For example, 13! is 6,227,020,800.

## How Factorials Are Used

Many mathematical formulas use factorial notation, including the formulas for finding **permutations** and **combinations** . For example, the number of permutations of *n* elements taken *n* at a time is *n* !, and the number of permutations of *n* elements taken *r* at a time is equal to .

There is also a problem that involves prime and **composite numbers** which uses a formula containing factorial notation. Mathematicians have, for many years, puzzled over the question of how **prime numbers** were distributed. Notice that, in the **whole numbers** less than 20, there are eight prime numbers (2, 3, 5, 7, 11, 13, 17, and 19). But from 20 to 40, there are only four prime numbers (23, 29, 31, and 37).

No one has yet found a formula that will generate all the prime numbers. However, the following sequence will give a string of *n* consecutive composite numbers (numbers that are *not* prime) for any positive integer *n.*

(*n* + 1)! + 2, (*n* + 1)! + 3, (*n* + 1)! + 4, (*n* + 1)! + 5, (*n* + 1)! + 6, and so on up to (*n* + 1)! + (*n* + 1).

When *n* is 2, notice that this sequence only has two terms:

(*n* + 1)! + 2, (*n* + 1)! + (*n* + 1)

which is

(2 + 1)! + 2, (2 + 1)! + (2 + 1)

For the first term, (2 + 1)! + 2 is 3! + 2 or (3 × 2 × 1) + 2, giving a value of 8. The second term has a value of 9.

When *n* = 2, this sequence gives two consecutive numbers that are *not* prime numbers: 8, 9. When *n* = 3, this sequence gives three consecutive numbers that are *not* prime numbers: 26, 27, 28. This relationship between the value of *n* and the number of consecutive numbers that are not prime numbers continues in this sequence for any whole number value for *n.* For a greater *n,* such as 300, a sequence of 300 composite numbers (that is, a list of 300 consecutive numbers with no prime number in the list) can be found.

see also Factors; Primes, Puzzles of; Permutations and Combinations.

*Lucia McKay*

## Bibliography

Stephens, Larry. *Algebra for the Utterly Confused.* New York: McGraw-Hill, 2000.

# Factorial

# Factorial

The number n! is the product 1 × 2 × 3 × 4 ×... × n, that is, the product of all the **natural numbers** from 1 up to n, including n itself where 1 is a natural number. It is called either "n factorial" or "factorial n." Thus 5! is the number 1 × 2 × 3 × 4 × 5, or 120.

Older books sometimes used the symbol In for n factorial, but the numeral followed by an exclamation **point** is currently the standard symbol.

Factorials show up in many formulas of **statistics** , probability, **combinatorics** , **calculus** , **algebra** , and elsewhere. For example, the formula for the number of permutations of n things, taken n at a **time** , is simply n!. If a singer chooses eight songs for his or her concert, these songs can be presented in 8!, or 40,320 different orders. Similarly the number of combinations of n things r at a time is n! divided by the product r!(n - r)!. Thus the number of different bridge hands that can be dealt is 52! divided by 13!39!. This happens to be a *very* large number.

When used in conjunction with other operations, as in the formula for combinations, the factorial **function** takes precedence over **addition** , **subtraction** , negation, **multiplication** , and **division** unless parentheses are used to indicate otherwise. Thus in the expression r!(n - r)!, the subtraction is done first because of the parentheses; then r! and (r - n)! are computed; then the results are multiplied.

As n! has been defined, 0! makes no sense. However, in many formulas, such as the one above, 0! can occur. If one uses this formula to compute the number of combinations of 6 things 6 at a time, the formula gives 6! divided by 6!0!. To make formulas like this work, mathematicians have decided to give 0! the value 1. When this is done, one gets 6!/6!, or 1, which is, of course, exactly the number of ways in which one can choose all six things.

As one substitutes increasingly large values for n, the value of n! increases very fast. Ten factorial is more than three million, and 70! is beyond the capacity of even those calculators which can represent numbers in scientific notation.

This is not necessarily a disadvantage. In the series representation of sine x, which is x/1! - x3/3! + x5/5! -..., the denominators get large so fast that very few terms of the series are needed to compute a good decimal **approximation** for a particular value of sine x.

# factorial

fac·to·ri·al / fakˈtôrēəl/ •
n. Math. the product of an integer and all the integers below it; e.g., factorial four (*4!*) is equal to 24. (Symbol: !) ∎ the product of a series of factors in an arithmetic progression.•
adj. chiefly Math. relating to a factor or such a product: *a factorial design.*DERIVATIVES: fac·to·ri·al·ly adv.

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