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Magazine Chapter 5: Problem 12
Evaluate each expression. $$ { }_{9} C_{4} $$
Short Answer
Expert verified
The value of \text{ }_{9}C_{4} is 126.
Step by step solution
01
Understanding the Notation
Recognize that \text{ }_{9} C_{4} denotes a combination, which tells us in how many ways we can choose 4 items from 9 items without regard to the order of selection. The general formula for a combination is given by:\[ _{n}C_{r} = \frac{n!}{r!(n-r)!} \]
02
Assign Values to Variables
Identify the values from the given problem where \( n = 9 \) and \( r = 4 \)
03
Substitute Values into the Formula
Substitute \( n = 9 \) and \( r = 4 \) into the combination formula:\[ _{9}C_{4} = \frac{9!}{4!(9-4)!} \]
04
Simplify the Factorials
Calculate the factorials:\[ 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]\[ 4! = 4 \times 3 \times 2 \times 1 \]\[ (9-4)! = 5! = 5 \times 4 \times 3 \times 2 \times 1 \]
05
Cancel Common Terms
Substitute these values back into the combination formula and simplify:\[ _{9}C_{4} = \frac{9 \times 8 \times 7 \times 6 \times 5!}{4! \times 5!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} \]
06
Compute the Result
Finally, compute the resulting value:\[ _{9}C_{4} = \frac{9 \times 8 \times 7 \times 6}{24} = \frac{3024}{24} = 126 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
combination formula
In statistics and combinatorial mathematics, a combination is a way of selecting items from a larger pool such that the order of selection does not matter. The combination formula is fundamental to solving these types of problems. It is given by: \[ _{n}C_{r} = \frac{n!}{r!(n-r)!} \] Here, \( n \) represents the total number of items, while \( r \) is the number of items to choose.
The formula utilizes factorials to calculate the number of possible combinations where the order of items is irrelevant. We often use this formula in probability and statistics to determine the number of ways a subset can be chosen from a larger set.
The formula utilizes factorials to calculate the number of possible combinations where the order of items is irrelevant. We often use this formula in probability and statistics to determine the number of ways a subset can be chosen from a larger set.
factorials
Factorials are key components in the combination formula. A factorial, denoted by \( n! \), is the product of all positive integers up to \( n \). For example:
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
- \( 0! = 1 \) (by definition)
binomial coefficient
The binomial coefficient is another name for combinations and is used in binomial expansions. It indicates the number of ways to choose \( r \) items from \( n \) items and is represented as \( _{n}C_{r} \). For example, \( _{9}C_{4} \) denotes choosing 4 items from 9.
This coefficient is found in the expansion of a binomial raised to a power, such as \[ (a + b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r} b^r \] Here, \( \binom{n}{r} \) represents the binomial coefficients. These coefficients can be calculated using the combination formula and are instrumental in combinatorial proofs and probability calculations.
This coefficient is found in the expansion of a binomial raised to a power, such as \[ (a + b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r} b^r \] Here, \( \binom{n}{r} \) represents the binomial coefficients. These coefficients can be calculated using the combination formula and are instrumental in combinatorial proofs and probability calculations.
combinatorial mathematics
Combinatorial mathematics deals with counting, arranging, and finding patterns among sets of items. It includes a variety of topics such as the study of combinations, permutations, and binomial coefficients.
Understanding combinatorial principles helps solve problems where the arrangement or selection of objects is important but where the order does not always matter.
Key concepts include:
Understanding combinatorial principles helps solve problems where the arrangement or selection of objects is important but where the order does not always matter.
Key concepts include:
- Permutations: Arrangements of items where order matters.
- Combinations: Selections of items where order does not matter.
- Factorials: Used in calculating permutations and combinations.
- Binomial coefficients: Count the ways to choose subsets of a set.
