Factorial Questions with Solutions

Tutorial on evaluating and simplifying expressions with factorial notation are presented.

Definition of Factorial

Question 1

1. \( 4! \)
   
2. \( 5! \times 5! \)
   
3. \( 3! \times 0! \)
   
4. \( \dfrac{ 4!}{0!} \)
   
5. \( \dfrac{6!}{2! \times 4!} \)

Solution to Question 1

1. \( 4! = 1 \times 2 \times 3 \times 4 = 24 \)
   
2. \( 5! \times 5! = (5!)^2 = (1 \times 2 \times 3 \times 4 \times 5)^2 = 120^2 = 14400 \)
   
3. \( 3! \times 0! = (1 \times 2 \times 3) \times 1 = 6 \)
   
4. \( \dfrac{ 4!}{0!} = \dfrac{1 \times 2 \times 3 \times 4}{1} = 24\)
   
5. \( \dfrac{6!}{2! \times 4!} \)
   \( = \dfrac{1 \times 2 \times 3 \times 4 \times 5 \times 6}{ (1 \times 2 ) \times ( 1 \times 2 \times 3 \times 4) } \)
   \( = 15 \)

Question 2

1. \( \dfrac{(n + 2)!}{n!} \)
   
2. \( \dfrac{(2n + 2)!}{(2n)!} \)
   
3. \( \dfrac{(n - 1)!}{(n + 1)!} \)

Solution to Question 2

1. Expand the factorials
   \( \dfrac{(n + 2)!}{n!} \)
   \( = \dfrac{ 1 \times 2 \times ... \times n \times (n + 1) \times (n + 2) } {1 \times 2 \times ...\times n } \)
   Cancel common factors, in the numerator and denominator, and simplify to obtain
   \( = (n + 1)(n + 2) \)
   
2. Expand the factorials
   \( \dfrac{(2n + 2)!}{(2n)!} \)
   \( = \dfrac{ 1 \times 2 \times 3...(2n) \times (2n + 1) \times (2n + 2) } {1 \times 2 \times 3...2n} \)
   Cancel common factors, in the numerator and denominator, and simplify to obtain
   \( = (2n + 1) \times (2n + 2) \)
   
3. Expand the factorials
   \( \dfrac{(n - 1)!}{(n + 1)!} \)
   \( = \dfrac {1 \times 2 \times 3...(n - 1) }{1 \times 2 \times 3...(n - 1) \times n \times (n + 1)} \)
   Cancel common factors in the numerator and denominator and simplify to obtain
   \( = \dfrac{1}{n(n+1)} \)

Exercises

Answers to above exercises

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