Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-22,\) evaluate each expression. $$ _{12} C_{7} $$

Step-by-Step Solution

Answer

The value of \( _{12}C_{7} \) is 792.

Step 1: Understanding Combinations

Combinations are ways of choosing items from a larger set where the order does not matter. The formula to calculate combinations is given by \[ _{n}C_{r} = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

Step 2: Setting Up the Formula

Substitute \( n = 12 \) and \( r = 7 \) into the combination formula: \[ _{12}C_{7} = \frac{12!}{7!(12-7)!} \] This simplifies to \[ _{12}C_{7} = \frac{12!}{7!5!} \]

Step 3: Calculate Factorials

Calculate the factorials involved in the formula. 1. \( 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7! \) 2. \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \) 3. \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \)

Step 4: Simplifying the Expression

Use the factorial calculations to simplify the expression: \[ _{12}C_{7} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7!}{7! \times 5!} \] Cancel out \( 7! \) from the numerator and the denominator.Now it becomes: \[ \frac{12 \times 11 \times 10 \times 9 \times 8}{5!} \]

Step 5: Calculate Final Result

Now calculate the product of the remaining numerator:\( 12 \times 11 = 132 \ 132 \times 10 = 1320 \ 1320 \times 9 = 11880 \ 11880 \times 8 = 95040 \)Then, calculate the denominator:\( 5! = 120 \)Finally, compute the combination:\[ _{12}C_{7} = \frac{95040}{120} = 792 \]

Key Concepts

Factorials

In mathematics, the concept of a factorial is fundamental when it comes to permutations and combinations. The factorial of a non-negative integer, denoted by an exclamation mark (!), is the product of all positive integers less than or equal to that number. For example, the factorial of 5, written as \(5!\), is calculated as \(5 \times 4 \times 3 \times 2 \times 1 = 120\). This concept helps in counting possibilities and is crucial for solving problems related to arrangement and selection.

Some key points about factorials:

• Factorial of 0 is defined as 1 (\(0! = 1\)). This is a special case that's important in combinatorial mathematics.
• Factorial grows rapidly with larger numbers. For instance, \(6! = 720\) and \(7! = 5,040\).
• Factorials are used in various fields like physics and statistics, especially in probability theory.

Understanding how to compute factorials is essential when dealing with formulas for permutations and combinations.

Permutation

Permutation refers to the arrangement of items in a specific order. Unlike combinations, where the order does not matter, in permutations, sequence is crucial. For example, the words 'dog' and 'god' are different permutations of the same set of letters.

The formula to calculate permutations when choosing \(r\) items from \(n\) distinct items is:\[_{n}P_{r} = \frac{n!}{(n-r)!}\]Here’s what this formula signifies:

• \(n\) is the total number of items available.
• \(r\) is the number of items to arrange.
• \(n!\) accounts for arranging all items, then dividing by \((n-r)!\) to account for items not chosen.

For example, if you want to find out how many ways you can arrange 3 out of 5 books on a shelf, you'd use the permutation formula: \(_{5}P_{3} = \frac{5!}{(5-3)!} = \frac{5!}{2!}\). Calculating this gives you 60 possible arrangements.

Binomial Coefficients

Binomial coefficients are used to determine the number of ways to choose \(r\) items from \(n\) items without regard to order. This concept is crucial in combinations and directly applies to problems such as the one given where \(_{12}C_{7}\) was calculated.

The binomial coefficient can be represented mathematically as:\[ _{n}C_{r} = \frac{n!}{r!(n-r)!}\]This formula involves:

• \(n!\) which represents the factorial of the total items.
• \(r!\) which is the factorial of the selected items.
• \((n-r)!\) accounting for the unselected items.

Binomial coefficients provide a way to calculate the number of combinations for selecting items, which is significant in probability and algebra, particularly when expanding expressions like \((a + b)^n\). Understanding and using binomial coefficients correctly ensures you can handle a wide range of combinatorial problems efficiently.