Problem 5 - Algebra and Trigonometry

Question

1–6 Evaluate the expression. $$P(100,1)$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Permutation Notation

The notation \( P(n, r) \) represents a permutation, which is a way of selecting \( r \) objects from \( n \) distinct objects where order matters.

Step 2: Apply the Permutation Formula

The formula for permutations is given by \( P(n, r) = \frac{n!}{(n-r)!} \). We need to substitute the given numbers into this formula.

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 Permutation Expression

The factorial expression \( \frac{100!}{99!} \) simplifies to 100 because all terms in the factorial cancel out except for 100. This is calculated as: \[ \frac{100 \times 99 \times 98 \times \ldots \times 1}{99 \times 98 \times \ldots \times 1} = 100 \].

Key Concepts

Factorial

A factorial, denoted as \(!\) is a mathematical operation that involves multiplying a series of descending natural numbers. When you see a number followed by an exclamation mark, such as \(n!\), it means:

• Multiply \(n\) by every positive integer below it.
• For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
• The definition for \(0!\) is 1. This might seem strange, but it is a necessary definition to make many mathematical formulas work properly.

Factorials grow extremely fast. Even relatively small numbers can lead to very large results. Understanding factorials is crucial for calculations involving permutations and combinations.

Permutation Formula

The permutation formula is a tool used in combinatorics to find the number of ways to arrange a set number of objects, where the order is important. The formula for permutations is given as: \[ P(n, r) = \frac{n!}{(n-r)!} \] Here:

• \(n\) is the total number of items to choose from.
• \(r\) is the number of items to arrange or permute.
• \((n-r)!\) accounts for the leftover items that don't need arranging.

To evaluate \(P(100, 1)\), plug in the values: \[ P(100, 1) = \frac{100!}{(100 - 1)!} = \frac{100!}{99!} \] When you simplify \(\frac{100!}{99!}\), all the numbers from 99 downwards cancel out, leaving you with 100. This simplification shows how effectively the permutation formula identifies the number of arrangements with order importance.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects following specific rules. It includes topics like:

• Permutations – where order matters.
• Combinations – where order doesn’t matter.

When working with permutations, as in our exercise, it's crucial to recognize that the sequence in which the objects are arranged can create a different outcome. Combinatorics helps in:

• Understanding how many ways you can choose and arrange objects.
• Simplifying complex counting problems using systematic counting strategies.
• Giving insights into probability and other areas in mathematics and science.

By practicing problems that use concepts from combinatorics, such as permutations, students can develop a better grasp of order and arrangement significance.