Problem 44 - Algebra 2

Question

Evaluate each expression. \(\frac{7 !}{3 !}\)

Step-by-Step Solution

Answer

The value of \(\frac{7!}{3!}\) is 840.

Step 1: Understand Factorials

In this exercise, you are asked to evaluate an expression involving factorials. Remember, a factorial, denoted with an exclamation mark (!), is the product of an integer and all the integers below it. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1\).

Step 2: Calculate 7!

Calculate the factorial of 7. This means multiplying all integers from 7 down to 1: \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\).

Step 3: Calculate 3!

Next, calculate the factorial of 3, which means multiplying all integers from 3 down to 1: \(3! = 3 \times 2 \times 1 = 6\).

Step 4: Divide 7! by 3!

Now, divide the result of \(7!\) by \(3!\): \(\frac{7!}{3!} = \frac{5040}{6}\).

Step 5: Simplify the Division

Perform the division to simplify the expression: \(\frac{5040}{6} = 840\).

Key Concepts

Integer Multiplication

Understanding integer multiplication is crucial when dealing with factorials. Factorials involve multiplying a sequence of descending integers in a particular order. For instance, in the expression \(7!\), you multiply 7 by every integer below it down to 1: \(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\).

• Order Matters: It’s important to multiply in sequence to ensure accuracy. The result is dependent on this order.
• Collective Product: The product grows rapidly because we multiply by another integer each time.

This concept is not just about speed or efficiency but about understanding that each multiplication adds to the complexity of the product. This principle is the foundation of calculating any factorial expression.

Simplifying Expressions

When faced with complex expressions, simplifying them makes it easier to solve. This is especially true with factorial expressions. For example, calculating the expression \(\frac{7!}{3!}\) can seem daunting because of large numbers, but we simplify by recognizing common factors.

• Cancellation: Factorials like \(3!\) are part of \(7!\), as \(3 \times 2 \times 1\) cancel out in both the numerator and denominator.
• Reduction: This subtraction helps reduce calculations. Since \(3!\) effectively cancels, you're left with multiplying only the factors from 7 down to 4 in \(7!\), because the rest are divided out.

Simplifying in this manner minimizes errors and reduces unnecessary calculation load, making solving such expressions more manageable.

Division of Factorials

The division of factorials involves understanding how to condense large numbers into simple fractions. When dividing \(7!\) by \(3!\), you simplify before calculating directly, which saves time and prevents errors.

• Use of Factorials: By recognizing that \(3!\) is part of \(7!\), because it’s the product of the smaller part of the sequence, you can directly cancel these terms.
• Final Simplification: After cancellation, you only need to calculate the remaining terms, \(7 \times 6 \times 5 \times 4\), giving \(840\).

This process of division by simplification is powerful because it leverages the inherent structure within the factorials, keeping computations straightforward without manual errors from managing extensive multiplication.