Problem 18 - Precalculus: Mathematics for Calculus

Question

Evaluate the expression. $$\left(\begin{array}{l}8 \\ 3\end{array}\right)$$

Step-by-Step Solution

Answer

The value of the expression is 56.

Step 1: Identify the Expression

The expression \(\binom{8}{3}\), read as '8 choose 3,' is a binomial coefficient. It represents the number of ways to choose 3 items from 8 without regard to the order.

Step 2: Write the Formula for Binomial Coefficient

The formula to calculate \(\binom{n}{r}\) is given by:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]where \(n!\) denotes the factorial of \(n\).

Step 3: Plug Values into the Formula

Using the formula from Step 2, plug in \(n = 8\) and \(r = 3\):\[\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!}\]

Step 4: Calculate Factorials

Compute the factorials needed:- \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320\) - \(3! = 3 \times 2 \times 1 = 6\)- \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)

Step 5: Simplify the Expression

Insert the calculated factorials back into the expression from Step 3:\[\binom{8}{3} = \frac{40320}{6 \times 120} = \frac{40320}{720}\]

Step 6: Compute the Final Result

Divide \(40320\) by \(720\):\[40320 \div 720 = 56\]Therefore, \(\binom{8}{3} = 56\).

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics and are used extensively in the calculation of binomial coefficients. The factorial of a non-negative integer, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). It's a way of representing permutations where order matters. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). This means there are 120 different ways to arrange 5 distinct items in order.

Key properties of factorials include:

• \( 0! = 1 \): This is a commonly agreed convention.
• Factorials grow very rapidly as the value of \( n \) increases.
• Used in combinations and permutations.

Factorials play a critical role in understanding combinatorics, as they provide the basis upon which combinations are derived.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations, arrangements, and counting. It's quintessential when solving problems related to choosing items from a larger set. In our current context, combinatorics helps us determine how many ways we can pick 3 items from a set of 8, without concern for the order they are picked in. This concept is explored through the binomial coefficient, also known as 'n choose r', where \( n \) is the total number of items, and \( r \) is the number of items to choose.

The formula is:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, the factorials help simplify the calculation of the number of combinations. By performing these calculations, problems involving selection without replacement or concern for order can easily be solved. Combinatorics is not only a theoretical framework but is applicable in real-world scenarios like lottery draws, probability theory, and statistical analysis.

Mathematical Notation

Mathematical notation is a universal language that describes mathematical ideas in a concise form. In the context of binomial coefficients, the notation \( \binom{8}{3} \) is used. This notation is read as '8 choose 3' and succinctly communicates the operation of selecting 3 objects from a group of 8.

Notation serves several purposes such as:

• Reducing complexity: It simplifies complex mathematical ideas into manageable expressions.
• Providing clarity: It makes mathematical communication precise and unambiguous.
• Facilitating calculation: By using standard symbols and expressions, calculations can be conducted and verified with efficiency.

The notation \( n! \) for factorials and \( \binom{n}{r} \) for combinations ensures that mathematical expressions convey exact meanings, allowing problems in algebra, calculus, and combinatorics to be tackled methodically and reliably. Understanding notation is key to mastering mathematical concepts and solving equations effectively.