Problem 31 - A Graphical Approach to College Algebra

Question

Use a calculator to evaluate each expression. $$_{15} C_{8}$$

Step-by-Step Solution

Answer

The value of \( _{15}C_{8} \) is 6435.

Step 1: Understand the Notation

The expression \( _{n}C_{r} \) represents a combination, which is the number of ways to choose \( r \) objects from \( n \) objects without regard to order. In this case, it is \( _{15}C_{8} \).

Step 2: Formula for Combinations

The formula for combinations is \( _{n}C_{r} = \frac{n!}{r!(n-r)!} \). For \( _{15}C_{8} \), we will use \( n = 15 \) and \( r = 8 \).

Step 3: Calculate Factorials

Calculate the factorials needed:- \( 15! = 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).- \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).- \( (15 - 8)! = 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).

Step 4: Plug into the Formula

Substitute the factorials into the formula:\[ _{15}C_{8} = \frac{15!}{8! \times 7!} \].

Step 5: Simplify the Expression

Cancel out the common terms in the numerator and the denominator:- In the numerator \( 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7! \).- In the denominator \( 8! \times 7! \).Thus, the expression simplifies to:\[ \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} \].

Step 6: Final Calculation using a Calculator

Use a calculator to compute \( \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9}{40320} = 6435 \).

Key Concepts

Factorials

Factorials are an essential concept when dealing with combinations and permutations. A factorial is represented by the symbol "!" and is defined as the product of all positive integers up to a given number. This means if you have a number, say 5, the factorial of 5 (written as 5!) is 5 \( \times \) 4 \( \times \) 3 \( \times \) 2 \( \times \) 1 = 120. The key features of factorials include:

• Multiplication sequence: It involves multiplying a series of descending natural numbers.
• Base case: The factorial of 0 is defined to be 1, i.e., 0! = 1.
• Role in formulas: Factorials appear in many mathematical formulas due to their usefulness in counting and permutations.

Factorials grow very quickly, making them vital yet challenging when working with large numbers. In the context of combinations, factorials help calculate the total possible ways to arrange or select items from a larger set.

Binomial Coefficient

The binomial coefficient is a key concept for determining combinations. It is usually represented as \( _{n}C_{r} \) or \( \binom{n}{r} \), and it tells you how many ways you can choose \( r \) objects from \( n \) without considering the order. Here’s how it works:

• Formula: The formula for the binomial coefficient is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).
• Used in combinations: This coefficient is central when calculating combinations, essentially providing the main calculation method.
• Symmetry: An interesting property of the binomial coefficient is that \( \binom{n}{r} = \binom{n}{n-r} \), which reveals its inherent symmetry.

In simple terms, the binomial coefficient helps you figure out the total number of ways to pick items from a larger group. It plays a fundamental role in statistical calculations and probability theory.

Permutations and Combinations

When dealing with sets, permutations and combinations are two different ways of counting the elements. While both involve the arrangement of objects from a set, the key difference lies in whether the order of arrangement matters. Let’s explore the basic concepts:

• Permutations: These are arrangements where the order matters. For example, the arrangement AB is different from BA.
• Combinations: These are selections where the order does not matter. For example, selecting AB is the same as selecting BA.
• Formulas: The number of permutations of \( n \) distinct objects is \( n! \). The number of combinations is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).

This distinction is crucial when solving problems involving selection and arrangement. In many real-world applications, such as cryptography and logistics, understanding these differences can be instrumental for designing efficient systems.