What does [tex]$\frac{3!}{3!}$[/tex] equal?
A. 1
B. [tex]$\frac{3 \times 2 \times 1}{2 \times 1}$[/tex]
C. 3
D. 6
Answer (1)
First, understand that the problem requires evaluating the expression 3 ! 3 ! .
Calculate 3 ! as 3 × 2 × 1 = 6 .
Substitute the calculated value into the expression, resulting in 6 6 .
Evaluate the fraction to find the final answer: 6 6 = 1 .
Explanation
Understanding the Problem We are asked to evaluate the expression 3 ! 3 ! . The factorial of a non-negative integer n , denoted by n ! , is the product of all positive integers less than or equal to n .
Calculating 3!
First, let's calculate 3 ! . By definition, 3 ! = 3 × 2 × 1 = 6 .
Substituting the Value Now, substitute the value of 3 ! into the expression: 3 ! 3 ! = 6 6 .
Evaluating the Fraction Finally, evaluate the fraction: 6 6 = 1 .
Final Answer Therefore, 3 ! 3 ! = 1 .
Examples
Understanding factorials and their ratios is useful in probability and combinatorics. For example, if you want to find the probability of arranging 3 distinct objects in a specific order out of all possible arrangements, you would use factorials. In this case, 3 ! 3 ! simplifies to 1, illustrating a basic concept in these fields.
