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Chapter 11: Problem 17

Evaluate each expression. $$\frac{5 !}{0 !}$$

Short Answer

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120

Step by step solution

01

Understand Factorials

Factorials are represented by an exclamation mark (!). For any positive integer n, the factorial n! is the product of all positive integers less than or equal to n. For example, 5! = 5 脳 4 脳 3 脳 2 脳 1.
02

Evaluate 5!

Calculate the factorial of 5. \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
03

Understand 0!

By definition, the factorial of 0 is 1. \( 0! = 1 \)
04

Divide 5! by 0!

Now divide the factorial of 5 by the factorial of 0. \( \frac{5!}{0!} = \frac{120}{1} = 120 \)

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Key Concepts

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

Factorial Notation
Factorial notation is a way of representing the product of a sequence of descending natural numbers. The notation involves an exclamation mark (!) after a number. When you see something like 5!, it means you need to multiply all whole numbers from 5 down to 1. So, 5! is equal to 5 脳 4 脳 3 脳 2 脳 1, which results in 120. Similarly, 4! is 4 脳 3 脳 2 脳 1, which equals 24. Factorials are extremely useful in various areas of mathematics, including permutations, combinations, and series expansions.

Understanding factorial notation helps simplify many mathematical expressions and problems. Always remember to multiply down to 1, and you can easily evaluate any factorial expression.
Factorial of Zero
The factorial of zero might seem confusing at first because there are no positive integers less than zero to multiply. By mathematical convention, 0! is defined to be 1. This definition maintains the consistency and usefulness of the factorial function across different formulas and mathematical contexts.

For example, the definition of combinations involves the factorial of numbers, including zero, so defining 0! as 1 ensures the formula for combinations works correctly, even when choosing zero elements. Practically, you will often see 0! in various equations, always remember that it equals 1.
Factorial Division
Factorial division involves dividing the factorial of one number by the factorial of another number. For instance, if you have \(\frac{5!}{2!}\), you first calculate each factorial separately: 5! = 120 and 2! = 2. Then, you simply divide one result by the other: \(\frac{5!}{2!} = \frac{120}{2} = 60\).

In our original problem, we needed to evaluate \(\frac{5!}{0!}\). We know that 5! equals 120 and 0! equals 1, so \(\frac{5!}{0!} = \frac{120}{1} = 120\).

With factorial division, knowing the basic values and properties of factorials, such as 0! being 1, helps you quickly find the solution to these types of problems.

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Most popular questions from this chapter

Consider the sample space of 36 equally likely outcomes to the experiment in which a pair of dice is rolled. In each case determine whether the events \(A\) and \(B\) are mutually exclusive. \(A:\) The sum is twelve. \(B:\) The numbers are different.

Write the complete binomial expansion for each of the following powers of a binomial. $$\left(2 a+b^{2}\right)^{3}$$

Use the addition rule to solve each problem. If \(P(E)=0.2, P(F)=0.7,\) and \(P(E \cup F)=0.8,\) then what is \(P(E \cap F)\) ?

Prove that \(\sum_{i=0}^{n}\left(\begin{array}{l}n \\\ i\end{array}\right)=2^{n}\) for any positive integer \(n\)

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