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Definition of Factorial
Let n be a positive integer. n factorial, written n!, is defined by
n! = 1 * 2 * 3 * ... (n - 1) * n
The special case when n = 0, 0 factorial is given by: 0! = 1
Question 1: Evaluate the following expressions:
- 4!
- 5! * 5!
- 3! * 0!
- 4! / 0!
- 6! / (2! * 4!)
Solution to Question 1:
- 4! = 1 * 2 * 3 * 4 = 24
- 5! * 5! = (1 * 2 * 3 * 4 * 5) 2 = 120 2 = 14400
- 3! * 0! = (1 * 2 * 3) * 1 = 6
- 4! / 0! = (1 * 2 * 3 * 4) / 1 = 24
- 6! / (2! * 4!)
= (1 * 2 * 3 * 4 * 5 * 6) / [ (1 * 2 ) * ( 1 * 2 * 3 * 4) ]
= 15
Question 2: Simplify the following expressions:
- (n + 2)! / n!
- (2n + 2)! / 2n!
- (n - 1)! / (n + 1)!
Solution to Question 2:
- Expand the factorials
(n + 2)! / n!
= [ 1 * 2 * ... * n * (n + 1) * (n + 2) ] / [ 1 * 2 * ...* n ]
and simplify to obtain
= (n + 1)(n + 2)
- Expand the factorials
(2n + 2)! / 2n!
= [ 1 * 2 * 3...(2n) * (2n + 1) * (2n + 2) ] / [ 1 * 2 * 3...2n ]
Simplify
= (2n + 1) * (2n + 2)
- Expand the factorials
(n - 1)! / (n + 1)!
= [ 1 * 2 * 3...(n - 1) ] / [ 1 * 2 * 3...(n - 1) * n * (n + 1) ]
Simplify
= 1 / [ n * (n + 1) ]
Exercises:
a) Evaluate (10! / 5!) / 10
b) Simplify (n + 1)! / n!
Answers to above exercises:
a) (10! / 5!) / 10 = 3024
b) (n + 1)! / n! = n + 1
More references on elementary statistics and probabilities.
Factorial Calculator. Calculate the factorial of a positive integer.
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