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Chapter 4: Problem 8

Evaluate \(9 !\)

Short Answer

Expert verified
The value of \(9!\) is 362880.

Step by step solution

01

Understanding the Problem

The task is to evaluate the factorial of 9, which is denoted as \(9!\). The factorial of a number \(n\) is the product of all positive integers less than or equal to \(n\). Therefore, we need to compute \(9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\).
02

Calculate Step by Step

Start with the largest number in the sequence and multiply sequentially: \[9 \times 8 = 72\]
03

Multiply Result with Next Factor

Continue multiplying with the next smaller integer: \[72 \times 7 = 504\]
04

Continue Multiplying

Multiply the result by the next integer in the sequence: \[504 \times 6 = 3024\]
05

Proceed with Multiplication

Multiply by the next integer: \[3024 \times 5 = 15120\]
06

Further Multiplication

Continue with the sequence: \[15120 \times 4 = 60480\]
07

Next Integer Multiplication

Multiply by the next integer: \[60480 \times 3 = 181440\]
08

Complete the Calculation

Multiply by the final integer: \[181440 \times 2 \times 1 = 362880\]
09

Verify the Result

Ensure that you have accounted for all integers from 9 down to 1, yielding the evaluated factorial of 9 as 362880.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mathematics Education
Understanding factorials, like evaluating (9!) , serves as a fundamental building block in mathematics education. When students learn about factorials, they engage with the core idea of multiplying a sequence of decreasing positive integers. These types of exercises are often incorporated into mathematics education to solidify students' understanding of multiplication sequences and promote number sense.
  • A key element of learning is grasping the concept that a factorial represents the total number of ways to arrange (n) distinct items, which links directly to combinatorics and permutations.
  • This concept fosters a deeper comprehension of multiplication, as well as factors, which students can build upon as they progress in their studies.
Educators aim to simplify complex mathematical concepts through clear examples and incremental challenges, enabling students to eventually tackle more advanced mathematical problems with confidence.
Problem-Solving Steps
Approaching mathematical problems systematically is essential for accurate calculations such as evaluating (9!) . Problem-solving steps help in ensuring that the task is manageable and that each part of the solution process is clearly understood.
  • Begin by clearly understanding the problem, as done in Step 1, where we define (9!) as the product of all positive integers less than or equal to 9.
  • Carry out the calculations in a stepwise manner, systematically addressing each multiplication: starting with the largest number and breaking it down, as seen in Step 2 through Step 8.
  • Finally, verification in Step 9 gives an opportunity to review the entire process and double-check the work done to ensure accuracy.
These steps encourage thoroughness and logical thinking, which are both invaluable skills in mathematics and problem-solving.
Number Theory
Number theory is an essential aspect of understanding how factorial calculations interplay with mathematical theory. Factorials like (9!) , involve a sequence of multiplications, shedding light on a number's structural properties.
  • Facilitating the understanding of prime numbers: Since factorials involve multiplying a sequence of integers, students can learn to spot factors and primes within these sequences.
  • Deepening the comprehension of divisibility rules: Factorials can be used to explore patterns of even and odd numbers, reinforcing basic number theory principles.
Factorial calculations expose students to practical applications of number theory, enhancing their ability to recognize and work with number-based patterns, a skill vital for advanced studies in mathematics and related fields.

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Most popular questions from this chapter

An experiment is designed to test the potency of a drug on 20 rats. Previous animal studies have shown that a \(10-\mathrm{mg}\) dose of the drug is lethal \(5 \%\) of the time within the first 4 hours; of the animals alive at 4 hours, \(10 \%\) will die in the next 4 hours. What is the probability that 2 rats will die in the 8-hour period?

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