Problem 9 - Algebra and Trigonometry

Question

Evaluate the expression. $$ P(100,1) $$

Step-by-Step Solution

Answer

The value of \( P(100, 1) \) is 100.

Step 1: Understand the Meaning of the Notation

The notation \( P(n,r) \) is used to represent the number of permutations of \( n \) items taken \( r \) at a time. Permutations consider the order of items, which means each arrangement is unique.

Step 2: Use the Permutation Formula

The formula for permutations \( P(n, r) \) is given by \( P(n, r) = \frac{n!}{(n-r)!} \). We need to use this formula to evaluate \( P(100, 1) \).

Step 3: Substitute the Values into the Formula

Substitute \( n = 100 \) and \( r = 1 \) into the permutation formula: \[P(100, 1) = \frac{100!}{(100-1)!} = \frac{100!}{99!}\]

Step 4: Simplify the Expression

Recognize that \( 100! = 100 \times 99! \), so the expression becomes: \[\frac{100 \times 99!}{99!}\]The \( 99! \) terms cancel out, leaving: \[100\]

Step 5: Conclude the Calculation

Based on the simplification, the value of the expression \( P(100, 1) \) is \( 100 \).

Key Concepts

Permutation Formula

Understanding permutations is crucial for solving problems where the order of selection matters. In mathematics, permutations can be used to count how many different ways we can order elements. The permutation formula is a handy tool to make these calculations manageable.

• The general formula for permutations when selecting \( r \) elements from \( n \) is given by \( P(n, r) = \frac{n!}{(n-r)!} \).
• This formula calculates how many ordered arrangements are possible.
• The notation \( P(n, r) \) refers to permutations of \( n \) items taken \( r \) at a time.

This formula is extremely useful in probability and combinatorics when we need to know how many different ways we can arrange or select a subset from a larger set. For instance, if you want to know how many ways you can arrange 100 books on a shelf in a row (1 at a time), \( P(100, 1) \), the permutation formula comes to the rescue. It's a powerful tool that helps compress complex counting into simple arithmetic.

Factorial

Factorials are at the heart of permutation calculations. The factorial of a number \( n \), denoted by \( n! \), is the product of all positive integers up to \( n \). This means:

• \( n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1 \)
• For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Factorials grow extremely fast with increasing \( n \). They are essential for permutations because they account for the different ways of rearranging a set of items. In permutations, when considering \( n \) objects arranged \( r \) at a time, the formula for permutation \( P(n, r) \) involves both \( n! \) and \( (n-r)! \). This is because we are interested in the number of ways of selecting \( r \) ordered elements from \( n \), not all \( n \) at once. Factorials help simplify the counting process in these instances, allowing us to use mathematical structure efficiently.

Arrangements

When we talk about arrangements, we're diving into how different objects can be ordered or sequenced. This is fundamentally what permutations deal with. Arrangements consider the sequence to be important, meaning two sequences that just swap two items are considered different.

• For example, arranging the letters A and B can result in AB or BA, which are two distinct arrangements.
• Permutations specifically give us a tool to count these sequences when the order is important, as opposed to combinations where order doesn’t matter.

The importance of arrangements becomes evident in practical scenarios like seating people in a row, ordering books on a shelf, or even arranging numbers in a specific sequence. In the problem of finding \( P(100, 1) \), we're simply arranging 1 item out of 100 possible ones, which intuitively results in 100 possible unique arrangements. This concept emphasizes that even when selecting just one item, it's crucial to consider that choosing any single item out of \( n \) gives us exactly \( n \) outcomes, reflecting all possible arrangements of single elements.