In the realm of mathematics, we encounter expressions that can challenge our problem-solving abilities and test our grasp of mathematical concepts. One such expression, presented as 4! • 3! / (12 • 30 • 144 • 5,040), might seem daunting at first glance. But fear not, for with a step-by-step approach and a solid understanding of factorials, we can confidently evaluate this expression and unveil its numerical answer.
In the realm of mathematics, we encounter expressions that can challenge our problem-solving abilities and test our understanding of fundamental concepts. One such expression is:
4! • 3! / (12)(30)(144)(5,040)
This expression combines factorials (denoted by the exclamation mark) and multiplication. While it may appear daunting at first glance, a step-by-step approach coupled with a solid grasp of factorials will pave the way to a clear and concise evaluation.
Understanding Factorials
A factorial, represented by the symbol n!, signifies the product of all positive integers less than or equal to n. For instance, 4! (pronounced “four factorial”) is equal to 4 x 3 x 2 x 1. Here’s a breakdown of factorials for the numbers used in the expression:
- 4! = 4 x 3 x 2 x 1 = 24
- 3! = 3 x 2 x 1 = 6
Deciphering the Expression
The expression presents a challenge because it combines factorials with direct multiplication. To conquer this, we’ll employ a strategic approach. We’ll first evaluate the factorials, then tackle the multiplication, and finally, perform the division.
Step 1: Evaluating the Factorials
As established earlier, 4! is 24 and 3! is 6. Let’s substitute these values into the expression:
24 • 6 / (12)(30)(144)(5,040)
Step 2: Performing the Multiplication
Now, we address the multiplication within the parentheses. Here, we can group factors with common multiples to simplify the process:
24 • 6 / (2 x 2 x 3)(2 x 3 x 5)(2 x 2 x 2 x 2 x 3 x 3)(2 x 2 x 2 x 3 x 5 x 7 x 8 x 9)
Multiplying each group of factors individually:
24 • 6 / (72)(30)(144)(2,880)
24 • 6 / 3,628,800
Step 3: Executing the Division
Finally, we perform the division:
24 • 6 / 3,628,800 = (24 x 6) / 3,628,800
Simplifying the division:
144 / 3,628,800
The Answer
After meticulously evaluating the expression, we arrive at the answer:
144 / 3,628,800 = 0.000 039 705 882 352 941 (approximately)
Key Points to Remember
- Factorials represent the product of positive integers less than or equal to a specified number.
- Evaluating expressions involving factorials often requires a step-by-step approach, prioritizing factorials before performing multiplication and division.
- When dealing with large numbers, calculators can be instrumental in ensuring accurate calculations.
Beyond the Calculation: Exploring Applications
While we’ve solved the expression, factorials and their applications extend far beyond this specific example. Factorials find significance in various mathematical disciplines, including:
- Combinatorics: Factorials play a crucial role in counting arrangements or selections of objects, forming the foundation for concepts like permutations and combinations.
- Probability Theory: Factorials are employed in calculating probabilities of events involving arrangements or selections.
- Statistics: Factorials surface in areas like calculating binomial coefficients, essential for statistical analysis.
- Calculus: Factorials serve as the building block for defining the derivative of the factorial function, a concept employed in advanced mathematical applications.
By grasping factorials and their applications, we unlock a deeper understanding of various mathematical concepts and enhance our problem-solving abilities.
Conclusion: The Power of Logical Evaluation
The expression we explored might have seemed daunting at first glance. However, by systematically dissecting it, evaluating factorials, and strategically performing calculations, we were able to reach the answer. This exercise serves as a testament to the power of logical evaluation and a sound understanding of mathematical concepts.
This approach can be applied to conquer even more complex expressions, fostering a deeper appreciation for the elegance and precision of mathematics.