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Chapter 5: Problem 23

Find the value of each combination. $$ { }_{52} C_{1} $$

Short Answer

Expert verified
The value of \({}_{52} C_1\) is 52.

Step by step solution

01

Understand the Combination Formula

The value of \({ }_{n} C_{r}\) (also written as \({n \choose r}\)) can be found using the formula: \[ { }_{n} C_{r} = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of items, \( r \) is the number of items to choose, and \( ! \) denotes factorial.
02

Substitute the Given Values

Here, \( n = 52 \) and \( r = 1 \). Substitute these values into the combination formula: \[ { }_{52} C_{1} = \frac{52!}{1!(52-1)!} \]
03

Simplify the Factorials

Simplify the factorials in the equation: \[ { }_{52} C_{1} = \frac{52!}{1! \cdot 51!} \]
04

Cancel Out Common Terms

Notice that \(52! = 52 \times 51!\), so the equation can be simplified by canceling out \(51!\): \[ { }_{52} C_{1} = \frac{52 \times 51!}{1! \times 51!} = \frac{52}{1} = 52 \]

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Key Concepts

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

combination formula
Combinations are a fundamental concept in statistics and probability. They're used when the order of selection does not matter. The combination formula is given by \[ { }_{n} C_{r} = \frac{n!}{r!(n-r)!} \] Here:
  • n represents the total number of items.
  • r represents the number of items to choose.
  • The exclamation mark (!) denotes a factorial.
By understanding this formula, you can determine the number of different ways to choose a subset of items from a larger set, without considering the order.
factorials
In mathematics, a factorial is a function that multiplies a number by every number below it. The factorial of a non-negative integer n is denoted by n! and is equal to: \[ n! = n \times (n-1) \times (n-2) \times \text{...} \times 3 \times 2 \times 1 \] For example:
  • 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
  • 3! = 3 \times 2 \times 1 = 6
Factorials are a key part of the combination formula because they help calculate how many ways items can be arranged or selected in a set. When simplifying, terms often cancel out, as seen in the example provided in the exercise.
binomial coefficients
Binomial coefficients are the numerical factors in the expansion of a binomial raised to a power, and they are represented using combinations. The binomial coefficient \(\binom{n}{r}\) is the same as the combination formula and is calculated by: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] These coefficients appear in the Binomial Theorem, which provides a way of expanding expressions that are raised to a power. For instance, the expression \((x + y)^n\) can be expanded using binomial coefficients. Understanding these coefficients is useful for solving problems involving combinations and probabilities.
probability
Probability measures how likely an event is to occur and is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. When dealing with combinations, probability often calculates the likelihood of selecting a particular subset from a larger set. For example:
  • The probability of drawing 1 specific card from a deck of 52 cards is \( P(A) = \frac{ { }_{52} C_{1} }{52} = \frac{52}{52} = 1 \)
Combining probabilities with combinations allows us to address more complex questions, like determining the likelihood of getting a certain hand in a game of cards. Remember, understanding the link between combinations and probability can greatly enhance your ability to solve statistical problems.

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Most popular questions from this chapter

A box containing twelve 40-watt light bulbs and eighteen 60-watt light bulbs is stored in your basement. Unfortunately, the box is stored in the dark and you need two 60-watt bulbs. What is the probability of randomly selecting two 60-watt bulbs from the box?

In a recent Harris Poll, a random sample of adult Americans (18 years and older) was asked, "When you see an ad emphasizing that a product is 'Made in America,' are you more likely to buy it, less likely to buy it, or neither more nor less likely to buy it?" The results of the survey, by age group, are presented in the following contingency table. $$ \begin{array}{lrrrrr} & \mathbf{1 8 - 3 4} & \mathbf{3 5 - 4 4} & \mathbf{4 5 - 5 4} & \mathbf{5 5 +} & \text { Total } \\ \hline \text { More likely } & 238 & 329 & 360 & 402 & \mathbf{1 3 2 9} \\ \hline \text { Less likely } & 22 & 6 & 22 & 16 & \mathbf{6 6} \\ \hline \begin{array}{l} \text { Neither more } \\ \text { nor less likely } \end{array} & 282 & 201 & 164 & 118 & \mathbf{7 6 5} \\ \hline \text { Total } & \mathbf{5 4 2} & \mathbf{5 3 6} & \mathbf{5 4 6} & \mathbf{5 3 6} & \mathbf{2 1 6 0} \end{array} $$ (a) What is the probability that a randomly selected individual is 35-44 years of age, given the individual is more likely to buy a product emphasized as "Made in America"? (b) What is the probability that a randomly selected individual is more likely to buy a product emphasized as "Made in America," given the individual is \(35-44\) years of age? (c) Are 18 - to 34 -year-olds more likely to buy a product emphasized as "Made in America" than individuals in general?

Suppose that Ralph gets a strike when bowling \(30 \%\) of the time. (a) What is the probability that Ralph gets two strikes in a row? (b) What is the probability that Ralph gets a turkey (three strikes in a row)? (c) When events are independent, their complements are independent as well. Use this result to determine the probability that Ralph gets a turkey, but fails to get a clover (four strikes in a row).

Go to www.pearsonhighered.com/sullivanstats to obtain the data file SullivanStatsSurveyI using the file format of your choice for the version of the text you are using. The data represent the results of a survey conducted by the author. The variable "Text while Driving" represents the response to the question, "Have you ever texted while driving?" The variable "Tickets" represents the response to the question, "How many speeding tickets have you received in the past 12 months?" Treat the individuals in the survey as a random sample of all U.S. drivers. (a) Build a contingency table treating "Text while Driving" as the row variable and "Tickets" as the column variable. (b) Determine the marginal relative frequency distribution for both the row and column variable. (c) What is the probability a randomly selected U.S. driver texts while driving? (d) What is the probability a randomly selected U.S. driver received three speeding tickets in the past 12 months? (e) What is the probability a randomly selected U.S. driver texts while driving or received three speeding tickets in the past 12 months?

For a parallel structure of identical components, the system can succeed if at least one of the components succeeds. Assume that components fail independently of each other and that each component has a 0.15 probability of failure. (a) Would it be unusual to observe one component fail? Two components? (b) What is the probability that a parallel structure with 2 identical components will succeed? (c) How many components would be needed in the structure so that the probability the system will succeed is greater than \(0.9999 ?\)
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