Problem 51 - Algebra for College Students

Question

Evaluate each expression without using a calculator. $$\frac{8 !}{3 ! 5 !}$$

Step-by-Step Solution

Answer

56 / 6 = 933.

Step 1: Understand the Factorial

The factorial of a number is the product of all positive integers up to that number. For example, \[ n! = n \times (n-1) \times (n-2) \times \text{...} \times 2 \times 1 \].

Step 2: Write Down the Factorials

List the factorials involved in the expression. So, \[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \], \[ 3! = 3 \times 2 \times 1 \], \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 \].

Step 3: Substitute the Factorials in the Expression

Replace the factorials in the expression with the expanded forms:\[ \frac{8!}{3! 5!} = \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1) ( 5 \times 4 \times 3 \times 2 \times 1)} \].

Step 4: Simplify the Expression

Cancel out the common terms in the numerator and the denominator. Since \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 \], we can cancel out \[5, 4, 3, 2, \text{and} 1 \] in the numerator and the denominator, resulting in:\[ \frac{8 \times 7 \times 6}{3 \times 2 \times 1} \].

Step 5: Simplify Further

Compute the remaining parts of the numerator and denominator separately. Simplify \[8 \times 7 \times 6 / 3 \times 2 \times 1\] by performing division step-by-step: \[ \frac{8}{1} \times \frac{7}{1} \times \frac{6}{6} = 1 \]. Hence, simplifying further:\[ 8 \times 7 / 6 = 8 \times 7 / 3 \times 2 \]. Finally:\[ \frac{56}{6} = \frac{56}{3 \times 2} \therefore 56 / 6 = 56/ 6\], simplifying to 56/6=933

Key Concepts

Factorials

Factorials are mathematical expressions that represent the product of all positive integers up to a given number. For instance, the factorial of 5, written as \(5!\), is \[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\].
Factorials are denoted by an exclamation mark (!). Here is the general formula for a factorial of a number \(n\): \[n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1\].
Understanding factorials is crucial because they are widely used in permutations, combinations, and other areas of algebra. When solving problems that include factorials, always start by expanding the factorial expressions to their basic multiplicative form.

Combinations

Combinations are a way to select items from a larger set where the order does not matter. The formula for combinations, often written using binomial coefficients, is: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\].
Here, \(n\) represents the total number of items, and \(k\) represents the number of items to choose. The concept of combinations is crucial in probability, statistics, and various algebra problems.
For example, if we need to find the number of ways to choose 3 items out of 8, we use \[\binom{8}{3} = \frac{8!}{3!5!}\].
By simplifying the factorials in the formula, we can solve the combination problem step-by-step.

Algebraic Simplification

Algebraic simplification involves reducing complex expressions into simpler forms. When simplifying expressions involving factorials, the key step is to cancel out common factors.
For instance, in the expression \[\frac{8!}{3!5!}, \]
we first expand the factorials: \[8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\], \[3! = 3 \times 2 \times 1\], and \[5! = 5 \times 4 \times 3 \times 2 \times 1\].
Next, we substitute the expanded factorials into the expression and cancel out the common terms: \[\frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1 \times 5 \times 4 \times 3 \times 2 \times 1}\].
After canceling, we have: \[\frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{56}{6} = 9.333\].

Binomial Coefficient

The binomial coefficient is a numeric value that appears in the binomial theorem and is used to calculate combinations. It is represented by \[\binom{n}{k}\] and reads as 'n choose k.'
It tells us how many ways we can choose \(k\) items from \(n\) without regard to order. The binomial coefficient is calculated using factorials:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\].
This coefficient is extremely useful for simplifying and solving problems that involve combinations and algebraic expressions, as seen in the exercise with \[\frac{8!}{3!5!}\].
Understanding the binomial coefficient helps in navigating complex algebraic problems efficiently.