Problem 30 - A Graphical Approach to College Algebra

Question

Use a calculator to evaluate each expression. $$100 C_{5}$$

Step-by-Step Solution

Answer

752875

Step 1: Understand the Combination Formula

The combination formula is used to calculate the number of ways to choose a subset of items from a larger set. The formula is given by \[ nCr = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of items, and \( r \) is the number of items to choose.

Step 2: Identify the Values of n and r

In our problem, \( n = 100 \) and \( r = 5 \). We need to evaluate \( 100 C_{5} \), which represents choosing 5 items from a set of 100.

Step 3: Apply the Values to the Formula

Substitute \( n = 100 \) and \( r = 5 \) into the combination formula:\[ 100 C_{5} = \frac{100!}{5!(100-5)!} \] This simplifies to:\[ 100 C_{5} = \frac{100!}{5! \cdot 95!} \]

Step 4: Calculate Each Factorial

Calculate the factorials: - \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)- Since \( 100! \) and \( 95! \) are very large, we utilize the expression:\[ \frac{100!}{95!} = 100 \times 99 \times 98 \times 97 \times 96 \]

Step 5: Evaluate the Expression

Compute \( \frac{100 \times 99 \times 98 \times 97 \times 96}{120} \).- First, calculate the numerator: \( 100 \times 99 \times 98 \times 97 \times 96 = 9034502400 \)- Then divide by the denominator, 120: \( \frac{9034502400}{120} = 752875 \)

Step 6: Interpret the Result

The value of \( 100 C_{5} \) is 752875. This means there are 752875 ways to choose 5 items from 100.

Key Concepts

Combination Formula

In combinatorics, the combination formula is essential for calculating how many different ways you can select a group of items from a larger pool without regard to the order of selection. It is distinct from permutations, where the order does matter. The combination formula is expressed as:\[ nCr = \frac{n!}{r!(n-r)!} \]Here, \( n \) represents the total number of items you have, and \( r \) is the number of items you want to choose. The formula divides the total arrangements, given by \( n! \), by the number of ways to arrange the \( r \) items, as well as the remaining \( n-r \) items. This ensures that only distinct combinations are counted.
For example, if you want to choose 5 items from a total of 100, the combination is denoted as \( 100 C_5 \). You will then apply the formula by setting \( n = 100 \) and \( r = 5 \), transforming it into \( \frac{100!}{5!(100-5)!} \), as described in the outlined exercise steps. Using this formula, you can find how many unique groups of 5 can be chosen.

Factorial Calculation

Factorial calculations are foundational in both permutations and combinations. The factorial of a positive integer, denoted by \( n! \), is the product of all positive integers up to \( n \). For instance, \( 5! \) means you multiply 5 by all the whole numbers below it, down to 1, which gives \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Factorials can grow very rapidly, especially with larger numbers. For instance, calculating \( 100! \) involves multiplying all integers from 1 to 100, which is impractical for direct calculation due to size. Fortunately, we can simplify many problems. In the case of computing combinations like \( 100 C_5 \), you can use the relation \( \frac{100!}{95!} = 100 \times 99 \times 98 \times 97 \times 96 \). Here, the large sections of the factorial calculation cancel out, significantly reducing the computation to manageable terms.

• Factorial notation: Facilitates concise expression of large products.
• Efficient simplification: Allows cancelation for more straightforward calculations.
• Practical computation: Makes use of cancellations to handle enormous numbers like \( 100! \).

Binomial Coefficient

The binomial coefficient, represented often as \( nCr \), is a key component in combinatorics and can be visually represented in Pascal's Triangle. It denotes the number of ways to choose \( r \) items from \( n \) items without considering the order. Within the framework of combinations, it is calculated using the combination formula.
For example, in the case of \( 100 C_5 \), the binomial coefficient shows how many unique selections of 5 items can be made from 100. These coefficients have widespread applications, not only in pure mathematics but also in practical realms like probability theory and statistics.

• Reflected in problems involving selection and arrangement.
• Empowers calculation of probabilities and complex statistical analysis.
• Fundamental in expanding binomial expressions and understanding distributions.

This mathematical tool, the binomial coefficient, enables both theoretical insights and practical solutions to myriad problems involving selection from larger sets.