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Chapter 5: Problem 24

Find the value of each combination. $$_{40} C_{40}$$

Short Answer

Expert verified
The value of \( _{40}C_{40} \) is 1.

Step by step solution

01

Understanding the Combination Formula

The combination formula is used to find the number of ways to choose items from a larger set without regard to the order of selection. The formula is given by \[ _n C_r = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of items, and \( r \) is the number of items to choose.
02

Substitute the Given Values

In this exercise, \( n = 40 \) and \( r = 40 \). Substitute these values into the formula: \[ _{40}C_{40} = \frac{40!}{40!(40-40)!} \] Simplify the equation inside the factorial: \[ _{40}C_{40} = \frac{40!}{40! \times 0!} \]
03

Simplify the Expression

Recall that \( 0! = 1 \). Thus, the equation simplifies to \[ _{40}C_{40} = \frac{40!}{40! \times 1} = \frac{40!}{40!} = 1 \]

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Key Concepts

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

combination formula
In probability and statistics, combinations are a way to select items from a larger set where the order of selection does not matter. This is different from permutations where order does matter. The combination formula is given by the expression: overtag{ _n C_r = \frac{n!}{r!(n-r)!} }where \( n \) represents the total number of items, and \( r \) is the number of items to choose from the total set. This formula helps in finding the number of possible ways to get a subset of items from a larger set.
factorial
Factorials are a key part of the combination formula. A factorial, denoted by \( n! \), is the product of all positive integers up to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials grow very quickly with larger values of \( n \), which makes them useful in combinatorial problems. An important point to remember is that by definition, \( 0! = 1 \). This special case is essential in combination calculations where \( n \) and \( r \) might be equal, as we saw in the exercise with \( _{40} C_{40} \).
binomial coefficient
The combination formula can also be referred to as the binomial coefficient. This term arises from its role in the binomial theorem, which provides a way to expand expressions of the form \( (a + b)^n \). The binomial coefficient, represented by \( \binom{n}{r} \), is the same as \( _n C_r \) and is calculated using the same factorial-based formula. In practice, the binomial coefficient gives the count of ways to pick \( r \) items from \( n \) without caring about the order of selection.
statistical methods
Combinations play a crucial role in various statistical methods. For example, they are used in hypothesis testing, where you might need to determine the number of possible outcomes. They are also important in probability calculations, like finding the likelihood of a certain number of successes in a sequence of independent events (binomial distribution). Understanding combinations, and by extension the combination formula, factorials, and the binomial coefficient, is fundamental to grasping more advanced statistical methods and concepts.

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Most popular questions from this chapter

Find the value of each combination. $$_{9} C_{2}$$

Packaging Error Through a manufacturing error, three cans marked "regular soda" were accidentally filled with diet soda and placed into a 12 -pack. Suppose that three cans are randomly selected from the 12 -pack. (a) Determine the probability that exactly two contain diet soda. (b) Determine the probability that exactly one contains diet soda. (c) Determine the probability that all three contain diet soda.

Clothing Options A man has six shirts and four ties. Assuming that they all match, how many different shirt-and-tie combinations can he wear?

The probability that a randomly selected individual in the United States 25 years and older has at least a bachelor's degree is \(0.272 .\) The probability that an individual in the United States 25 years and older has at least a bachelor's degree, given that the individual is Hispanic, is 0.114. Are the events "bachelor's degree" and "Hispanic" independent? (Source: Educational Attainment in the United States, 2003. U.S. Census Bureau, June 2004 )

Determine the probability that at least 2 people in a room of 10 people share the same birthday, ignoring leap years and assuming each birthday is equally likely by answering the following questions: (a) Compute the probability that 10 people have different birthdays. (Hint: The first person's birthday can occur 365 ways; the second person's birthday can occur 364 ways, because he or she cannot have the same birthday as the first person; the third person's birthday can occur 363 ways, because he or she cannot have the same birthday as the first or second person; and so on.) (b) The complement of "10 people have different birthdays" is "at least 2 share a birthday." Use this information to compute the probability that at least 2 people out of 10 share the same birthday.
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