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Chapter 11: Problem 5

Evaluate each expression. $$\frac{5 !}{3 ! 2 !}$$

Short Answer

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Step by step solution

01

- Understand Factorials

Recall that a factorial of a positive integer n is the product of all positive integers less than or equal to n. It is denoted by \(n!\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1\).
02

- Compute the Factorials

Calculate the factorials in the given expression: \[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\] \[3! = 3 \times 2 \times 1 = 6\] \[2! = 2 \times 1 = 2\]
03

- Apply Factorials to the Fraction

Substitute the calculated values back into the expression: \[\frac{5!}{3! 2!} = \frac{120}{6 \times 2} = \frac{120}{12}\]
04

- Simplify the Fraction

Now, simplify \(\frac{120}{12}\): \[\frac{120}{12} = 10\]

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Key Concepts

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

Factorial Definition
A factorial is a mathematical function applied to positive integers. It is written as \(n!\) and represents the product of all integers from 1 to n. For instance, if you want to find \(4!\), you multiply all numbers from 1 to 4 together: \(4! = 4 \times 3 \times 2 \times 1 = 24\). This principle is key in many areas of math, particularly in permutations and combinations. Whenever you see a factorial in a problem, remember it means multiplying in this way.
Simplifying Fractions
When simplifying fractions, the goal is to reduce them to their simplest form. This means dividing both the numerator and the denominator by their greatest common divisor. Let's consider the fraction \( \frac{120}{12}\). To reduce it, we divide 120 by 12 to get 10. Therefore, \( \frac{120}{12} = 10\). By simplifying fractions, you make calculations easier and results clearer. Always look for factors common to both the numerator and the denominator to streamline your work.
Combinatorics
Combinatorics is a branch of mathematics dealing with combinations, permutations, and the counting of objects. It's useful for solving problems where you need to know how many ways you can arrange or choose items. For example, \( \frac{5!}{3! 2!}\) is a combinatorial expression used in calculating combinations. It helps you determine the number of ways to choose a subset from a larger set. Understanding factorials and simplification is crucial here. They simplify calculations, making it easier to solve combinatorial problems efficiently.

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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.

Explain the difference between probability and odds.

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 \(P(E \cap F)\) ?

Determine the foci, asymptotes, and intercepts for the hyperbola \(\frac{x^{2}}{25}-\frac{y^{2}}{36}=1\)

Write the complete binomial expansion for each of the following powers of a binomial. $$(x+3)^{3}$$
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