Problem 52 - Algebra for College Students

Question

Evaluate each expression without using a calculator. $$\frac{8 !}{2 ! 6 !}$$

Step-by-Step Solution

Answer

28

Step 1: Identify the factorial expressions

The given expression is \(\frac{8!}{2! 6!}\). Factorial notation \(!\) means the product of all positive integers up to that number. For example, \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\).

Step 2: Write out the factorials

Write out the factorials for 8, 2, and 6. \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\), \(2! = 2 \times 1\), and \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\).

Step 3: Simplify the factorial expression

Notice that \(6!\) appears in the numerator and the denominator. Therefore, cancel out \(6!\) from both parts of the expression: \(\frac{8 \times 7 \times 6!}{2! \times 6!} = \frac{8 \times 7}{2!}\).

Step 4: Simplify the remaining expression

Simplify \(\frac{8 \times 7}{2!}\). First compute \(2!\): \(2! = 2 \times 1 = 2\). Then the expression becomes \(\frac{8 \times 7}{2}\).

Step 5: Perform the final calculation

Finally, divide \(8 \times 7\) by 2: \( \frac{56}{2} = 28 \). Thus, the value of the expression is 28.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics concerned with counting, arranging, and finding patterns in sets of objects. In this exercise, we use factorials, which are common in combinatorics. Factorials help compute permutations and combinations, which represent the different ways of arranging or choosing items from a set.
For example, the expression \(\frac{8!}{2! 6!}\) in combinatorics represents finding the number of ways to choose 2 items from a set of 8, denoted as _C(8,2)_. Factoring down the expression simplifies the solution of such problems.
Understanding where to apply combinatorics in real-life examples, like seating arrangements or selecting committee members, will deepen your comprehension of these principles.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form to make calculations easier. In this problem, we simplified \(\frac{8!}{2! 6!} \) step by step.
1. First, recognize that both the numerator and the denominator contain factorials. Factorials are products of consecutive positive integers.
2. Next, note any common factors in the expressions. Here, \(6!\) appears in both the numerator and the denominator, and thus can be canceled out.
3. Lastly, simplify what's left. Ensure you perform the arithmetic operations correctly to avoid mistakes. Breaking down each part of an expression to see which components can be eliminated makes the process manageable and reduces complexity.

Basic Arithmetic

Basic arithmetic underpins all mathematics, involving the simple operations of addition, subtraction, multiplication, and division. Let’s review how these are applied step by step in the context of this problem:
1. Multiplication of a sequence, as with \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\).
2. Performing division accurately, such as simplifying \( \frac{56}{2}\) to get our final solution.

Arithmetic ensures the foundation for all higher-level math. By understanding and practicing these fundamental operations, complex problems become much simpler and clearer.