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 on
elementary statistics and probabilities.

Factorial Calculator. Calculate the factorial of a positive integer.

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