Evaluate the expression. [tex]$\frac{91}{3!}$[/tex]
Answer (1)
First, calculate 3 ! = 3 × 2 × 1 = 6 .
Then, divide 91 by 6: 6 91 = 15.166666666666666 .
The final result is 15.166666666666666 .
Explanation
Understanding the expression We are asked to evaluate the expression 3 ! 91 . To do this, we first need to calculate the value of 3 ! .
Calculating the factorial The factorial of a number n , denoted by n ! , is the product of all positive integers less than or equal to n . Therefore, 3 ! = 3 × 2 × 1 = 6 .
Substituting the factorial value Now we can substitute this value back into the original expression: 3 ! 91 = 6 91 .
Performing the division To find the value of 6 91 , we perform the division. The result is 15.166666... , which can also be expressed as the mixed number 15 6 1 .
Final Answer Therefore, the value of the expression 3 ! 91 is 15.166666666666666 .
Examples
Understanding factorials and how to evaluate expressions involving them is crucial in many areas, such as probability and combinatorics. For example, if you want to calculate the probability of arranging 3 books on a shelf in a specific order out of a set of 91 books, you would need to understand factorials. The total number of ways to arrange 3 books out of 91 is given by ( 91 − 3 )! 91 ! = 91 × 90 × 89 , which involves understanding how factorials work. Simplifying expressions with factorials helps in determining the number of possible arrangements or combinations in various scenarios.
