Question

Evaluate each factorial expression. \(\frac{12 !}{10 !}\)

Short answer

132

Step-by-step solution

Write Down the Factorial Expressions

First, write down the given expression in terms of factorials: \(\frac{12!}{10!}\).

Expand the Factorials

Expand the factorial expressions. Recall that \(n!\) means the product of all positive integers up to \(n\). So, we have: \(12! = 12 \times 11 \times 10!\) and \(10! = 10!\).

Simplify the Fraction

Substitute the expanded factorials back into the expression to get: \(\frac{12 \times 11 \times 10!}{10!}\). We observe that \(10!\) in the numerator and denominator cancel out. Therefore, we are left with: \(12 \times 11\).

Calculate the Result

Multiply the remaining numbers: \(12 \times 11 = 132\).

Key concepts

Factorial Expression

Factorial expressions involve the product of all positive integers up to a specified number. This concept is essential in many mathematical and statistical applications. For instance, if we have 5!, it means we multiply all integers from 1 up to 5. In mathematical terms, that’s \(5! = 1 \times 2 \times 3 \times 4 \times 5 = 120\). This applies to any factorial expression \(n!\), where \(n\) is a positive integer.

Simplifying Fractions

Simplifying fractions makes the expressions easier to handle and reduces the complexity of calculations. In our example expression, \(\frac{12!}{10!}\), simplifying fractions involves canceling out common terms in the numerator and denominator.
By recognizing that both the numerator \(12! = 12 \times 11 \times 10!\) and the denominator \(10!\) contain \(10!\), we can cancel these terms. This leaves us with a simplified fraction \(\frac{12 \times 11 \times 10!}{10!} = 12 \times 11\).

• This step is pivotal as it brings us closer to our solution without over-complicating the calculation process.

Mathematical Operations

Various mathematical operations simplify and solve expressions like factorials. These basic operations include addition, subtraction, multiplication, and division. Understanding how to apply these correctly can help solve complex problems easily.
In our factorial example, multiplication is the key operation used to simplify \(\frac{12!}{10!}\). Recognizing and correctly performing these operations is a fundamental skill in mathematics.
Start by breaking down each part of the expression systematically. Write down the expanded form, cancel out common terms, and then perform the necessary multiplication.

Integer Multiplication

Once we have simplified the factorial expression, the final step is integer multiplication. This is straightforward if we follow the correct steps.
In our exercise, after breaking down the factorial and canceling out the common terms, we are left with \(12 \times 11\). Multiplying these two integers together: \(12 \times 11 = 132\).

• Integer multiplication involves multiplying two whole numbers without fractions or decimals.
• If you struggle with multiplying larger integers, breaking them down into smaller, more manageable parts can help.
• Remember, practice is key to mastering these fundamental mathematical skills.