Evaluate the expression. [tex]$4! \cdot 3!$[/tex]
Answer (2)
Calculate 4 ! = 4 × 3 × 2 × 1 = 24 .
Calculate 3 ! = 3 × 2 × 1 = 6 .
Multiply the results: 24 × 6 = 144 .
The final answer is 144 .
Explanation
Understanding the problem We are asked to evaluate the expression 4 ! × 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 4!
First, let's calculate 4 ! . By definition, 4 ! = 4 × 3 × 2 × 1 = 24 .
Calculating 3!
Next, let's calculate 3 ! . By definition, 3 ! = 3 × 2 × 1 = 6 .
Multiplying the results Now, we multiply the results: 4 ! × 3 ! = 24 × 6 = 144 .
Final Answer Therefore, the value of the expression 4 ! × 3 ! is 144 .
Examples
Factorials are used in probability and combinatorics to count the number of ways to arrange items. For example, if you have 4 books, 4 ! tells you how many ways you can arrange them on a shelf. If you then have 3 CDs, 3 ! tells you how many ways you can arrange them. Multiplying these together, 4 ! ⋅ 3 ! , gives the total number of ways to arrange the books and CDs together, assuming the order of books and CDs matters.
The expression 4 ! ⋅ 3 ! evaluates to 144, calculated by first determining the values of 4 ! and 3 ! , which are 24 and 6, respectively. The product of these two results gives 144 .
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