Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-22,\) evaluate each expression. $$ _{12} C_{5} $$

Step-by-Step Solution

Answer

The value of \(_{12} C_{5}\) is 792.

Step 1: Understand the Notation

The expression \(_{12} C_{5}\) represents a binomial coefficient, also known as a combination. It is used to find the number of ways to choose 5 items from 12 without regard to order.

Step 2: Recall the Formula for Combinations

The formula for combinations is \( _{n} C_{r} = \frac{n!}{r! (n-r)!} \). Here, \(n=12\) and \(r=5\). Substitute these values into the formula.

Step 3: Calculate the Factorials

Compute \(12!\), \(5!\), and \( (12-5)! = 7!\). Factorials are the product of all positive integers up to that number.- \(12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7!\)- \(5! = 5 \times 4 \times 3 \times 2 \times 1\)- \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\)

Step 4: Simplify the Expression

Plug the factorials into the combination formula and simplify:\[_{12} C_{5} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} = \frac{95040}{120}\]

Step 5: Complete the Calculation

Perform the division:\[_{12} C_{5} = \frac{95040}{120} = 792\]

Key Concepts

Binomial Coefficient

Binomial coefficients are essential when dealing with combinations in mathematics, especially when selecting a subset of items from a larger set. The notation \(_{n} C_{r}\) is commonly used to represent the number of ways to choose \(r\) elements from a total of \(n\) elements. The key feature of combinations, as denoted by binomial coefficients, is that the order in which the items are selected does not matter. Hence, this is often referred to as a "combination without repetition."To find the value of a binomial coefficient, we use the formula\[_{n} C_{r} = \frac{n!}{r!(n-r)!}\]This formula tells us how many unique sets of \(r\) elements can be formed from \(n\) elements by considering the different ways to arrange \(n\) objects and accounting for the repetitions by dividing by the product of the factorials of \(r\) and \(n-r\). Understanding binomial coefficients is crucial in fields such as probability and statistics, where unpredictably drawing samples from a large population is standard practice.

Factorials

Factorials are a mathematical operation represented by \(!\) symbol. When calculating a factorial, you are finding the product of all positive integers up to a specific number. For example:

• \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)
• \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\)

Factorials grow very quickly with larger numbers, which makes them both powerful and sometimes a bit challenging to work with manually when solving problems.The factorial function is fundamental in combinatorics. It serves as the backbone in calculating permutations and combinations. When used in binomial coefficients, factorials help determine the total number of combinations of elements available, despite large numbers of possibilities.

Permutations

Permutations are similar to combinations but with a significant difference: order matters. While combinations are concerned only with the selection of items, permutations focus on the arrangement of those items.To calculate the permutations of choosing \(r\) items from a set of \(n\) items, the formula is given by:\[_{n} P_{r} = \frac{n!}{(n-r)!}\]This formula accounts for the order by considering how elements are arranged. Since permutations concern themselves with different orderings, the count can be significantly higher than combinations when dealing with the same dataset.Understanding permutations is crucial in problems where sequences and order play a critical role, such as scheduling, arranging items, and cryptography. It is the difference in methodology between permutations and combinations that highlights the importance of order in mathematical calculations.