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Chapter 11: Problem 14

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

Short Answer

Expert verified
The resultant of \(\frac{12 !}{10 !}\) is 132.

Step by step solution

01

Understanding Factorial Division

When dividing factorials, it is possible to cancel out common terms of the numerator and the denominator. Here, both 12! and 10! share a range of factorial components (10*9*8...1).
02

Simplify the Factorials

Specifically, \(12! = 12 * 11 * 10 * 9 * ... * 1\) and \(10! = 10 * 9 * ... * 1\). As the two share \(10 * 9 * ... * 1\), this can be cancelled from both providing: \(\frac{12 !}{10 !} = 12 * 11.\)
03

Evaluate the Resultant Expression

Now, with \( \frac{12 !}{10 !} = 12 * 11\), compute the product of 12 and 11 to get 132. Therefore, \(\frac{12!}{10!} = 132.\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factorial Division
Factorial division involves dividing one factorial by another. A factorial, denoted by the "!" symbol, is the product of all positive integers up to a given number. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1\).
In a division of factorials, like \(\frac{12!}{10!}\), we can cancel out the common terms from both the numerator and the denominator. This simplifies the calculation immensely.
Both 12! and 10! share the multiplied sequence from 10 down to 1, noted as \(10 \times 9 \times ... \times 1\).
  • The common part \(10!\) cancels out from both the numerator and the denominator.
  • This simplification leaves us with only the unique terms in the numerator, resulting in \(12 \times 11\).
Simplifying Factorials
Simplifying factorial expressions is all about reducing them by canceling identical sequences in both the numerator and the denominator. In our example, simplifying \(\frac{12!}{10!}\) hinges on recognizing overlapping elements.
The definition of each factorial shows us the overlap:
  • 12! = \(12 \times 11 \times 10 \times 9 \times ... \times 1\)
  • 10! = \(10 \times 9 \times ... \times 1\)
Finding the common sequence, which is 10!, is key to simplification. When you cancel this sequence from both the top and the bottom, it reduces the expression to its simplest form, \(12 \times 11\). Always look for these common sequences to make calculations simpler and efficient.
Evaluating Factorials
Once you've simplified a factorial expression, such as reducing \(\frac{12!}{10!}\) to \(12 \times 11\), the next step is to evaluate it.
Evaluating is essentially calculating the numerical value. For our simplified expression, multiply the remaining terms:
  • Multiply 12 by 11 to get \(132\).
This final computation gives the exact value of the factorial division, which is \(132\) in this case. Evaluating follows naturally after simplifying as you handle only the essential parts of the original factorial expression.

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Most popular questions from this chapter

In Exercises 11-14, a single die is rolled twice. Find the probability of rolling an odd number the first time and a number less than 3 the second time.

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