15 Aug, 2024
· Mathematics

How to evaluate the expression 4! • 3!

  • 12
  • 30
  • 144
  • 5,040
Short Answer
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Long Explanation

Explanation

Evaluating the Expression

To evaluate the expression 4!⋅3!4! \cdot 3!:

Factorial Definition

The factorial of a non-negative integer nn is defined as:

n!=n⋅(n−1)⋅(n−2)⋅…⋅2⋅1n! = n \cdot (n-1) \cdot (n-2) \cdot \ldots \cdot 2 \cdot 1

Calculating 4!4!

For 4!4!:

4!=4⋅3⋅2⋅1=244! = 4 \cdot 3 \cdot 2 \cdot 1 = 24

Calculating 3!3!

For 3!3!:

3!=3⋅2⋅1=63! = 3 \cdot 2 \cdot 1 = 6

Multiplying the Factorials

Now, multiply the two factorials:

4!⋅3!=24⋅64! \cdot 3! = 24 \cdot 6

Perform the multiplication:

4!⋅3!=1444! \cdot 3! = 144

Conclusion

Thus, the value of 4!⋅3!4! \cdot 3! is 144\mathbf{144}.

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Emily Rosen

Mathematics Content Writer at Math AI

Emily Rosen is a recent graduate with a Master's in Mathematics from the University of Otago. She has been tutoring math students and working as a part-time contract writer for the past three years. She is passionate about helping students overcome their fear of math through easily digestible content.

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Concept

Factorial Definition

Explanation

The factorial of a non-negative integer nn is the product of all positive integers less than or equal to nn. It is denoted by n!n!.

Definition in Formal Terms

The factorial of nn is mathematically defined as:

n!={1if n=0n×(n−1)×(n−2)×⋯×1if n>0n! = \begin{cases} 1 & \text{if } n = 0 \\ n \times (n-1) \times (n-2) \times \cdots \times 1 & \text{if } n > 0 \end{cases}

Examples

  • For n=0n = 0:
0!=10! = 1
  • For n=5n = 5:
5!=5×4×3×2×1=1205! = 5 \times 4 \times 3 \times 2 \times 1 = 120

Properties

  1. Recursive Property:
    • n!n! can be expressed recursively as:
n!=n×(n−1)!n! = n \times (n-1)!
  1. Growth:
    • Factorials grow very quickly with increasing nn. Even for small values like n=10n = 10, the result is quite large (10!=3,628,80010! = 3,628,800).

Usage

Factorials are used in various fields such as combinatorics, algebra, and analysis. They are essential in computing permutations and combinations, as well as in series expansions like Taylor series.

Concept

Calculating Individual Factorials

Calculating Individual Factorials

A factorial, denoted as n!n!, is a mathematical operation that multiplies a given number nn by all positive integers less than itself. This concept is fundamental in combinatorics and various fields of mathematics and computer science. The factorial of a non-negative integer nn can be defined recursively or iteratively.

Definition

Formally, the factorial of nn is given by:

n!={1if n=0n×(n−1)×(n−2)×⋯×1if n>0n! = \begin{cases} 1 & \text{if } n = 0 \\ n \times (n-1) \times (n-2) \times \cdots \times 1 & \text{if } n > 0 \end{cases}

Examples

  • The factorial of 5 (5!5!) is:
5!=5×4×3×2×1=1205! = 5 \times 4 \times 3 \times 2 \times 1 = 120
  • The factorial of 0 (0!0!) is defined as 1 by convention.

Properties

  1. Growth Rate: Factorials grow exponentially. For large nn, n!n! becomes very large, very quickly.

  2. Recursive Definition:

    n!=n×(n−1)!n! = n \times (n-1)!

Usage in Permutations and Combinations

Factorials are crucial in calculating permutations and combinations in probability and statistics. For instance, the number of ways to arrange nn distinct objects is n!n!.

Implementation

Recursively:

factorial(n)={1if n=0n×factorial(n−1)if n>0\text{factorial}(n) = \begin{cases} 1 & \text{if } n = 0 \\ n \times \text{factorial}(n-1) & \text{if } n > 0 \end{cases}

Iteratively:

factorial=1for i in range(1, n+1): factorial∗=i\text{factorial} = 1 \\ \text{for } i \text{ in range(1, n+1)}: \\ \text{ factorial} *= i

Understanding and calculating factorials is essential for solving many mathematical problems and performing specific computational tasks efficiently.