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.

Factorial

How Factorials Are Used

Bibliography