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Chapter 14: Problem 17

Evaluate each expression. $$ 4 ! \cdot 5 $$

Short Answer

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Step by step solution

01

Understand the Factorial

The factorial of a number is the product of all positive integers up to that number. For any positive integer n, it is denoted by n!. For example, the factorial of 4 is calculated as follows: \[ 4! = 4 \times 3 \times 2 \times 1 \]
02

Calculate 4!

Now that we understand what a factorial is, calculate 4!: \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]
03

Multiplication by 5

Next, multiply the result of 4! by 5: \[ 24 \times 5 = 120 \]

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Key Concepts

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

Multiplication
Multiplication is one of the basic operations in arithmetic. It involves combining groups of equal sizes. For example, if we have 4 groups of 3 apples each, we can use multiplication to quickly find out that there are 12 apples in total (4 * 3 = 12).

Multiplication can also be considered repeated addition. In the example above, instead of adding 3 four times (3 + 3 + 3 + 3), we can multiply 3 by 4. The result is the same, but it is a much quicker method.

When working with multiplication in more complex tasks like factorials, it is essential to understand that each step still follows the same principles of multiplication.
Positive integers
Positive integers are the set of all whole numbers greater than zero. These include 1, 2, 3, 4, and so on. They do not include negative numbers, fractions, or decimals.

Positive integers are crucial in many mathematical operations, including the calculation of factorials. When we say we are calculating the factorial of a number, we are specifically referring to these positive integers.

For example, the factorial operation involves multiplying a series of positive integers, starting from 1 up to the given positive integer (e.g., 4! involves multiplying 4, 3, 2, and 1).

Understanding positive integers is fundamental, as many mathematical procedures involve these numbers.
Factorial calculation
A factorial calculation finds the product of all positive integers up to a specified number. It is denoted by an exclamation mark (e.g., 4! means the factorial of 4).

Here鈥檚 how you calculate the factorial of 4 (4!):
  • Start with the number 4.
  • Multiply it by the next lowest positive integer: 4 * 3.
  • Continue multiplying by the next lower integers: (4 * 3) * 2 and then (4 * 3 * 2) * 1.
  • The calculation looks like this: 4! = 4 * 3 * 2 * 1, which equals 24.

Factorial calculations are very useful in various fields of mathematics, including combinatorics and probability.

In the exercise, after calculating the factorial of 4 (resulting in 24), you multiply that by 5 to get the final answer: 24 * 5 = 120.

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Most popular questions from this chapter

Find the indicated term for each sequence. $$ a_{n}=\frac{5 n-9}{3 n+8} ; \quad a_{16} $$

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