A factorial of n is the product of all integers from 1 to n, inclusive. The exclamation mark (!) is the unary operator that represents factorial. The expression '5!' is read, "Five factorial" and means 1·2·3·4·5 which equals 120. Zero factorial (0!) is treated as a special case and is defined to be 0! = 1. At this level of mathematics, the n in n! must be a non-negative integer.
In advanced mathematics, factorial is defined using the
gamma function:
Equation 1 |
n | Formula | n! |
---|---|---|
0 | None | 1 |
1 | 1! = 1 | 1 |
2 | 2! = 1·2 | 2 |
3 | 3! = 1·2·3 | 6 |
4 | 4! = 1·2·3·4 | 24 |
5 | 5! = 1·2·3·4·5 | 120 |
6 | 6! = 1·2·3·4·5·6 | 720 |
7 | 7! = 1·2·3·4·5·6·7 | 5040 |
8 | 8! = 1·2·3·4·5·6·7·8 | 40,320 |
9 | 9! = 1·2·3·4·5·6·7·8·9 | 362,880 |
10 | 10! = 1·2·3·4·5·6·7·8·9·10 | 3,628,800 |
Table 1 |
Since factorials get very large very quickly (see table 1), they can be difficult to calculate without a calculator. However, when used in mathematics, equations using factorials usually involve ratios of factorials. Since factorials by definition have many common factors, these problems can be reduced in difficulty by canceling common factors.
To understand this, start with the definition of a
combination:
Equation 1 |
If one wishes to calculate , the formula is: . |
First, write out the factorials: . |
Now use the property of multiplying by 1 to cancel all ones: . |
Now cancel the factors that are the same: . |
Now look for remaining factors that have common factors. In the case, 2·3 = 6 so the 6 on the top cancels the 2·3 on the bottom: . |
This has now become a simple multiplication problem. Since , there are exactly 56 ways that five objects selected from a pool of eight objects can be arranged. So, . |
# | A | B | C | D |
E | F | G | H | I |
J | K | L | M | N |
O | P | Q | R | S |
T | U | V | W | X |
Y | Z |
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