Factorial Questions with Solutions

Tutorial on evaluating and simplifying expressions with factorial notation.

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:
  1. 4!
  2. 5! × 5!
  3. 3! × 0!
  4. 4! / 0!
  5. 6! / (2! × 4!)

Solution to Question 1


  1. 4! = 1 × 2 × 3 × 4 = 24
  2. 5! × 5! = (1 × 2 × 3 × 4 × 5) 2 = 120 2 = 14400
  3. 3! × 0! = (1 × 2 × 3) × 1 = 6
  4. 4! / 0! = (1 × 2 × 3 × 4) / 1 = 24
  5. 6! / (2! × 4!)
    = (1 × 2 × 3 × 4 × 5 × 6) / [ (1 × 2 ) × ( 1 × 2 × 3 × 4) ]
    = 15

Question 2

Simplify the following expressions:
  1. (n + 2)! / n!
  2. (2n + 2)! / 2n!
  3. (n - 1)! / (n + 1)!

Solution to Question 2


  1. Expand the factorials
    (n + 2)! / n!
    = [ 1 × 2 × ... × n × (n + 1) × (n + 2) ] / [ 1 × 2 × ...× n ]
    and simplify to obtain
    = (n + 1)(n + 2)
  2. Expand the factorials
    (2n + 2)! / 2n!
    = [ 1 × 2 × 3...(2n) × (2n + 1) × (2n + 2) ] / [ 1 × 2 × 3...2n ]
    Simplify
    = (2n + 1) × (2n + 2)
  3. 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 and links

elementary statistics and probabilities.
Factorial Calculator to calculate the factorial of a positive integer.
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