Factorial Questions with Solutions
Tutorial on evaluating and simplifying expressions with factorial notation are presented.
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Definition of Factorial
Let \( n \) be a positive integer. \(n\) factorial , written \( n! \) , is defined by
\[ n! = 1 \times 2 \times 3 \times ... (n - 1) \times n \]
The special case when \( n = 0 \), \[ 0! = 1 \]
Question 1
Evaluate the following expressions:
- \( 4! \)
- \( 5! \times 5! \)
- \( 3! \times 0! \)
- \( \dfrac{ 4!}{0!} \)
- \( \dfrac{6!}{2! \times 4!} \)
Solution to Question 1
- \( 4! = 1 \times 2 \times 3 \times 4 = 24 \)
- \( 5! \times 5! = (5!)^2 = (1 \times 2 \times 3 \times 4 \times 5)^2 = 120^2 = 14400 \)
- \( 3! \times 0! = (1 \times 2 \times 3) \times 1 = 6 \)
- \( \dfrac{ 4!}{0!} = \dfrac{1 \times 2 \times 3 \times 4}{1} = 24\)
- \( \dfrac{6!}{2! \times 4!} \)
\( = \dfrac{1 \times 2 \times 3 \times 4 \times 5 \times 6}{ (1 \times 2 ) \times ( 1 \times 2 \times 3 \times 4) } \)
\( = 15 \)
Question 2
Simplify the following expressions:
- \( \dfrac{(n + 2)!}{n!} \)
- \( \dfrac{(2n + 2)!}{(2n)!} \)
- \( \dfrac{(n - 1)!}{(n + 1)!} \)
Solution to Question 2
-
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
\( = (n + 1)(n + 2) \)
-
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
\( = (2n + 1) \times (2n + 2) \)
-
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
a) Evaluate \( \dfrac{(10! / 5!)}{10} \)
b) Simplify \( \dfrac{(n + 1)!}{n!} \)
Answers to above exercises
a) \( \dfrac{(10! / 5!)}{10} = 3024 \)
b) \( \dfrac{(n + 1)!}{n!} = n +1 \)
More References and links
elementary statistics and probabilities .
Factorial Calculator to calculate the factorial of a positive integer.
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