Problem 26 - College Algebra

Question

Evaluate each factorial expression. $$\frac{20 !}{2!18 !}$$

Step-by-Step Solution

Answer

The evaluated factorial expression \(\frac{20 !}{2!18 !}\) equals to 190

Step 1: Simplify Factorial:

First, we write out the factorials in expanded form. So, \(20! = 20*19*18*17*16*....*3*2*1\), \(2! = 2*1\) and \(18! = 18*17*16*...*3*2*1\). By substituting these into the given expression, we get:\[\frac{20!}{2!*18!} = \frac{20*19*18*17*16*....*3*2*1}{(2*1)*(18*17*16*...*3*2*1)}\]

Step 2: Simplify the Expression:

Next, we can see that in the numerator and the denominator, the terms from 18 to 1 are common and therefore cancels out. This leaves us with:\[\frac{20*19}{2*1}\]

Step 3: Final Computation:

After performing the cancellations and multiplications, we get:\[\frac{20*19}{2*1} = 190\]

Key Concepts

Simplifying Factorials

When it comes to simplifying factorials, the process begins with understanding what a factorial really is. A factorial, denoted by an exclamation point (!), represents the multiplication of all positive integers up to that number. For example, \[n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1\]
This is a rather hefty operation, especially with larger numbers.

• In our example, we have \(20!\) which is equal to \(20 \times 19 \times 18 \times \ldots \times 3 \times 2 \times 1\).
• Further, \(2!\) is simply \(2 \times 1\), and \(18!\) is \(18 \times 17 \times \ldots \times 3 \times 2 \times 1\).

The goal here is not to perform all these multiplications, especially not manually. Instead, understanding that common parts of the numerator and the denominator may be used later to simplify the calculation saves time and effort. You will see how this works in the next step.

Cancellation in Fractions

Cancellation in fractions is a key technique in making calculations easier, especially when working with factorial expressions.

When you have a fraction, such as our given \(\frac{20!}{2!18!}\), you can cancel out the common factors in both the numerator and the denominator. This process simplifies the expression without changing its value.

• In our problem, observe that \(20!\) expands to \(20 \times 19 \times 18!\).
• Similarly, \(18!\) is present in both the numerator and the denominator and thus cancels out accordingly.
• After these cancellations, you are left with \(\frac{20 \times 19}{2 \times 1}\), which is much simpler to compute.

This step of cancellation drastically reduces the computational load, often turning seemingly complex calculations into much simpler arithmetic.

Combinatorial Expressions

Factorial expressions frequently appear in combinatorics, which is a field of mathematics dealing with combinations and arrangements. Whether calculating permutations or combinations, factorials play a pivotal role.Let's relate it to our exercise. By simplifying \(\frac{20!}{2!18!}\), we have essentially calculated a combination.

• This specific expression is known as "20 choose 2," written as \(\binom{20}{2}\).
• It represents the number of ways to choose 2 items from a set of 20, without considering the order of selection.

Using algebraic expressions like this one, you can solve real-world problems involving selection and arrangement. Being able to simplify them correctly by applying the concepts we've discussed also enhances your efficient problem-solving abilities in combinatorics.