Problem 20 - Precalculus Mathematics for Calculus

Question

Evaluate the expression. $$\left(\begin{array}{c}10 \\\5\end{array}\right)$$

Step-by-Step Solution

Answer

The value of \( \left(\begin{array}{c}10 \\ 5\end{array}\right) \) is 252.

Step 1: Understanding Combinations

The problem requires calculating the binomial coefficient, which counts the number of ways to choose a subset of 5 elements from a total of 10 elements without regard to the order of selection. This is denoted by \( \binom{10}{5} \).

Step 2: Using the Formula

The binomial coefficient is calculated using the formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Here, \( n = 10 \) and \( r = 5 \), so the expression becomes \( \binom{10}{5} = \frac{10!}{5!(10-5)!} \).

Step 3: Calculating Factorials

First, calculate the necessary factorials. \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \). For \( 5! \), it is \( 5 \times 4 \times 3 \times 2 \times 1 \) and for \( (10-5)! = 5! \), it is the same, so \( 5! = 120 \).

Step 4: Simplifying the Expression

Plug the values back into the formula: \( \binom{10}{5} = \frac{10!}{5!5!} \). Simplifying, we have: \( \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = \frac{30240}{120} = 252 \).

Step 5: Verification

Verify the calculation to ensure correctness. Multiply the terms in the numerator and divide by 120, confirming that \( \binom{10}{5} = 252 \).

Key Concepts

Factorials

Factorials are a simple yet fundamental concept in mathematics, especially when dealing with permutations and combinations. A factorial of a non-negative integer n, denoted as \( n! \), is the product of all positive integers less than or equal to n. For instance, the factorial of 4 is computed as \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). Factorials grow very quickly in value as the integer increases.

Key points about factorials include:

• 0! is defined to be 1.
• Factorials are used to measure the number of permutations of a set. For example, the number of ways to arrange 5 items is 5!, which equals 120.
• Factorials are essential in computing combinations and binomial coefficients.

Understanding factorials is crucial because they form the building blocks for more advanced concepts in combinatorics, like permutations and combinations. In our exercise, we use factorials to compute the binomial coefficient \( \binom{10}{5} \) with the \( nCr \) formula.

Combinations

Combinations are a way to calculate how many different groups can be formed from a larger set, where order does not matter. This is different from permutations, where the sequence of elements does matter. If you have 10 different books and want to select 5 to take on a trip without caring about the order, combinations help you count the number of possible selections.

For example, with a set of five letters, {A, B, C, D, E}, choosing three letters out of these gives combinations like {A, B, C} and {C, D, E}, where each different grouping of three letters is distinct.

The formula to calculate a combination is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where:

• \( n \) is the total number of items.
• \( r \) is the number of items to be chosen.

This formula, as seen in the exercise, allows us to find \( \binom{10}{5} \) which equals 252. This represents the number of distinct ways to select 5 items from a set of 10.

nCr Formula

The \( nCr \) formula is a notation used in mathematics for combinations, specifically referring to how many combinations of r items can be formed from a set of n items. It's expressed as \( \binom{n}{r} \), which we read as "n choose r." The formula for calculating this combination is:\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]This formula underscores the idea that we're choosing r items from a collection of n items without regard to the order of the items. To break it down:

• The numerator, \( n! \), accounts for the permutations of n items.
• The denominator, \( r!(n-r)! \), adjusts for the permutations within the smaller group of chosen items \( r \) and the ignored items \( n-r \).

The exercise is a perfect example of applying the \( nCr \) formula to solve for \( \binom{10}{5} \). By inserting the values for n=10 and r=5, we used this formula to find that the binomial coefficient is 252. This calculation highlights how useful and powerful the \( nCr \) formula is in determining various potential groupings from a larger set.