Problem 15
Question
Evaluate each expression. Do not use a calculator. $$P(9,2)$$
Step-by-Step Solution
Verified Answer
72
1Step 1: Understanding Permutations
The notation \( P(n, r) \) represents the number of permutations of \( n \) items taken \( r \) at a time. It is calculated using the formula \( P(n, r) = \frac{n!}{(n-r)!} \). In this problem, \( n = 9 \) and \( r = 2 \).
2Step 2: Substituting Values into the Formula
Substitute \( n = 9 \) and \( r = 2 \) into the formula for permutations: \( P(9, 2) = \frac{9!}{(9-2)!} = \frac{9!}{7!} \).
3Step 3: Simplifying the Expression
Now, simplify \( \frac{9!}{7!} \). Write 9 factorial as \( 9! = 9 \times 8 \times 7! \). Substitute this into the formula: \( \frac{9 \times 8 \times 7!}{7!} \).
4Step 4: Cancelling Common Factors
The \( 7! \) in the numerator and denominator cancels out, leaving \( 9 \times 8 \).
5Step 5: Multiplying Final Numbers
Finally, multiply the remaining numbers: \( 9 \times 8 = 72 \).
Key Concepts
FactorialnPr FormulaCombinatorics
Factorial
The concept of factorial is central in mathematics, especially when dealing with permutations and combinations. A factorial of a non-negative integer, denoted by the symbol
(!), is the product of all positive integers up to that number. For instance, the factorial of 5, written as 5!, is calculated as 5 × 4 × 3 × 2 × 1, which equals 120.
The factorial operation grows rapidly as the numbers increase. This is because each additional integer in the sequence is multiplied by all the preceding integers.
Factorials are very useful in permutations and combinations as they help calculate the number of ways to arrange or choose items. Understanding how to break down and simplify factorial expressions is key when solving permutation problems.
In our example, we focused on 9!, which we partially simplified since we only needed part of the factorial to solve for the permutation. Always remember, in permutation calculations, we often cancel out common factorial terms to simplify expressions!
The factorial operation grows rapidly as the numbers increase. This is because each additional integer in the sequence is multiplied by all the preceding integers.
Factorials are very useful in permutations and combinations as they help calculate the number of ways to arrange or choose items. Understanding how to break down and simplify factorial expressions is key when solving permutation problems.
In our example, we focused on 9!, which we partially simplified since we only needed part of the factorial to solve for the permutation. Always remember, in permutation calculations, we often cancel out common factorial terms to simplify expressions!
nPr Formula
The nPr formula is used to calculate permutations, which are arrangements of items in specific orders. The notation for permutations is given as \( P(n, r) \), where \( n \) is the total number of items you have, and \( r \) is the number of items you want to arrange.
The formula for finding permutations is:\[P(n, r) = \frac{n!}{(n-r)!}\]
This formula essentially divides the factorial of the total items by the factorial of the remaining items after choosing your 'r' items.
In our example, \( P(9, 2) \) means we want to know the number of ways to arrange two items out of nine. We calculated it as \( P(9, 2) = \frac{9!}{(9-2)!} \) which simplifies down because \( 9! = 9 × 8 × 7! \) and \( 7! \) cancels out, leaving us with \( 9 × 8 \). This represents the number of permutations of two items from a set of nine.
The nPr formula is critical for understanding how different elements can be taken and arranged when the order of placement matters!
The formula for finding permutations is:\[P(n, r) = \frac{n!}{(n-r)!}\]
This formula essentially divides the factorial of the total items by the factorial of the remaining items after choosing your 'r' items.
In our example, \( P(9, 2) \) means we want to know the number of ways to arrange two items out of nine. We calculated it as \( P(9, 2) = \frac{9!}{(9-2)!} \) which simplifies down because \( 9! = 9 × 8 × 7! \) and \( 7! \) cancels out, leaving us with \( 9 × 8 \). This represents the number of permutations of two items from a set of nine.
The nPr formula is critical for understanding how different elements can be taken and arranged when the order of placement matters!
Combinatorics
Combinatorics is a fascinating branch of mathematics that deals with counting, arranging, and grouping objects. It encompasses various principles including permutations and combinations.
Permutations, as we've seen, involve arranging objects in specific orders. Combinations, on the other hand, refer to selecting a group of items where the order does not matter. Both concepts answer different kinds of questions and are calculated using different formulas.
The study of combinatorics is essential because it provides techniques that are applicable in probability, computer science, and many areas of decision-making. When dealing with problems like ours, combinatorics help simplify the process of figuring out possible outcomes by using mathematical principles like the factorial to calculate arrangements and choices efficiently.
Understanding combinatorics is crucial, as it allows us to navigate through complexities inherent in real-world problems involving arrangement and selection, equipping us with problem-solving skills that extend beyond theoretical concepts.
Permutations, as we've seen, involve arranging objects in specific orders. Combinations, on the other hand, refer to selecting a group of items where the order does not matter. Both concepts answer different kinds of questions and are calculated using different formulas.
The study of combinatorics is essential because it provides techniques that are applicable in probability, computer science, and many areas of decision-making. When dealing with problems like ours, combinatorics help simplify the process of figuring out possible outcomes by using mathematical principles like the factorial to calculate arrangements and choices efficiently.
Understanding combinatorics is crucial, as it allows us to navigate through complexities inherent in real-world problems involving arrangement and selection, equipping us with problem-solving skills that extend beyond theoretical concepts.
Other exercises in this chapter
Problem 15
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