/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 28 Evaluate each expression. $$ ... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 28

Evaluate each expression. $$ \frac{_{5} C_{1} \cdot_{7} C_{2}}{_{12} C_{3}} $$

Short Answer

Expert verified
The expression is evaluated to be approximately 0.4773

Step by step solution

01

Calculate \(_{5} C_{1}\)

The combination formula is given as: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] where \( n \) is the total number of items, and \( k \) is the number of items to choose. So, \(_{5} C_{1} = \frac{5!}{1!(5-1)!} = 5\)
02

Calculate \(_{7} C_{2}\)

By applying the same formula as in Step 1, \(_{7} C_{2} = \frac{7!}{2!(7-2)!} = 21\)
03

Calculate \(_{12} C_{3}\)

Again applying the same formula, \(_{12} C_{3} = \frac{12!}{3!(12-3)!} = 220\)
04

Evaluate the Final Expression

Now, substitute the values from Step 1, Step 2 and Step 3 into the expression and calculate: \[\frac{_{5} C_{1} \cdot_{7} C_{2}}{_{12} C_{3}} = \frac{5 \cdot 21}{220} = \frac{105}{220} = 0.4773 \]

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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 helpful mathematical concept when working with combinations, permutations, and various calculations involving sequences. It is denoted by an exclamation mark
  • For any positive integer \(n\), the factorial is written as \(n!\), and it means the product of all positive integers less than or equal to \(n\).
  • For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
When using the factorial in combination problems, it helps in calculating the number of ways to choose items from a set. Factorials grow very quickly; for instance, while \(3!\) equals 6, \(10!\) escalates to a whopping 3,628,800.
Knowing how to handle factorials efficiently is crucial when working with the combination formula, which is used to find binomial coefficients.
Combination Formula
The combination formula is a mathematical tool used to determine the number of ways a subset can be selected from a larger set, where the order of selection does not matter.
It is often denoted as \(C(n, k)\), \(nCk\), or \(_{n}C_{k}\).
  • The formula itself is: \[C(n, k) = \frac{n!}{k!(n-k)!}\]
  • Here, \(n\) represents the total number of items in the set, and \(k\) is the number of items to choose.
This formula is built on factorial notation, and it is essential for solving problems like example expressions where we are combining more than one of these calculations.
Using this formula makes it easy to evaluate terms in binomial expansions and other statistical calculations quickly and accurately.
Binomial Coefficient
The binomial coefficient is a central concept in combinatorics and is closely related to the combination formula. It represents the number of ways to choose \(k\) elements from a set of \(n\) elements.
  • In simpler terms, it's a way to describe how to pick items from a set without considering the order.
  • It is expressed as \(\binom{n}{k}\) or \(C(n, k)\).
The binomial coefficient is a key building block in the binomial theorem, which describes the algebraic expansion of powers of a binomial.
By understanding the binomial coefficient, students can easily solve problems involving choosing subsets from a larger set, as seen in the example exercise. These coefficients also appear in various mathematical contexts, such as probabilities and statistical distributions.

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Most popular questions from this chapter

Find the term in the expansion of \(\left(x^{2}+y^{2}\right)^{5}\) containing \(x^{4}\) as a factor.

A degree-day is a unit used to measure the fuel requirements of buildings. By definition, each degree that the average daily temperature is below \(65^{\circ} \mathrm{F}\) is 1 degree-day. For example, an average daily temperature of \(42^{\circ} \mathrm{F}\) constitutes 23 degree-days. If the average temperature on January 1 was \(42^{\circ} \mathrm{F}\) and fell \(2^{\circ} \mathrm{F}\) for each subsequent day up to and including January 10 , how many degree-days are included from January 1 to January \(10 ?\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ 10-5+\frac{5}{2}-\frac{5}{4}+\dots-\frac{10}{1-\frac{1}{2}} $$

Use the formula \(a_{n}-4+(n-1)(-7)\) to find the eighth term of the sequence \(4,-3,-10, \ldots\)

Use the formula for \(_{n} C_{r}\) to solve To win at LOTTO in the state of Florida, one must correctly select 6 numbers from a collection of 53 numbers ( 1 through 53). The order in which the selection is made does not matter. How many different selections are possible?
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