Factorial of an integer n (written as n! and read as "n factorial") is simply a product of the number "n" and all natural numbers smaller than n. That is,
n! = n × (n-1) × (n-2) × … × 3 × 2 × 1
Note : Remember that factorial of 0 is 1. (0! = 1)
5! = 5 × 4 × 3 × 2 × 1
5!= 120
Note : 5! can also be written as 5 × 4!
Therefore, n! = n × (n-1)!
| n | n! | n (n-1)! |
|---|---|---|
| 1 | 1 | = 1 x 0! |
| 2 | 2 x 1 | = 2 x 1! |
| 3 | 3 x 2 x 1 | = 3 x 2! |
| 4 | 4 x 3 x 2 x 1 | = 4 x 3! |
| 5 | 5 x 4 x 3 x 2 x 1 | = 5 x 4! |
| 6 | 6 x 5 x 4 x 3 x 2 x 1 | = 6 x 5! |
Using the rule: n! = n × (n-1)!
11! = 11 × 10!
11! =11 × 3,628,800
11! = 39,916,800
Factorials are most commonly used in "Permutations and Combinations", a branch of Mathematics usually studied under "Probability". However, other areas of mathematics use factorials as well.
Permutations and Combinations are represented by following expression :
` (m!)/(n!)`
Where both m, n are integers.
`(12!)/(8!) ` = `(12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1)/(8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) ` …(1)
After cancelling like terms in the numerator and the denominator:
`(12!)/(8!)` = `(12 × 11 × 10 × 9)` = `11,880 `
Note from equation (1) that:
`(12!)/(8!)` = `(12×11×10×9×8!)/(8!)`
Factorials grow exponentially. It is said that in some cases, the growth of factorials is faster than that of exponentials.
| n | n! |
|---|---|
| 1 | 1 |
| 2 | 1 |
| 2 | 2 |
| 3 | 6 |
| 4 | 24 |
| 5 | 120 |
| 6 | 720 |
| 7 | 5,040 |
| 8 | 40,320 |
| 9 | 362,880 |
| 10 | 3,628,800 |
| 11 | 39,916,800 |
| 12 | 479,001,600 |
| 13 | 6,227,020,800 |
| 14 | 87,178,291,200 |
| 15 | 1,307,674,368,000 |
| 16 | 20,922,789,888,000 |
| 17 | 355,687,428,096,000 |
| 18 | 6,402,373,705,728,000 |
| 19 | 121,645,100,408,832,000 |
| 20 | 2,432,902,008,176,640,000 |
| 21 | 51,090,942,171,709,440,000 |
| 22 | 1,124,000,727,777,607,680,000 |
| 23 | 25,852,016,738,884,976,640,000 |
| 24 | 620,448,401,733,239,439,360,000 |
| 25 | 15,511,210,043,330,985,984,000,000 |
© 2023 iPracticeMath | All Rights Reserved | Terms of Use.