Multiplying and Dividing Factorials - Algebra II

Simplifying this equation we notice that the 3's, 2's, and 1's cancel so

Alternative Solution

This is a factorial question. The formula for factorials is .

The cancels out all of except for the parts higher than 4, this leaves a 6 and a 5 left to multilpy

The factorial sign (!) just tells us to multiply that number by every integer that leads up to it. So, can also be written as:

To make this easier for ourselves, we can cancel out the numbers that appear on both the top and bottom:

Recall that .

Likewise, .

Thus, the expression can be simplified in two parts:

and

The product of these two expressions is the final answer:

To simplify this, just write out each factorial:

This expression of factorials reduces to (n+1)(n+2). Therefore, the solution must be a number that multiplies to 2 consecutive integers. Only 30 is a product of 2 consecutive integers.

So n would have to be 4 in this problem.

A factorial is a number which is the product of itself and all integers before it. For example

In our case we are asked to divide by . To do this we will set up the following:

We know that can be rewritten as the product of itself and all integers before it or:

Substituting this equivalency in and simplifying the term, we get:

Remember what a factorial is, and first write out what the original equation means. A factorial is a number that you multiply by all whole numbers that come before it until you reach one.

You can simplify because all terms in the expression 17! are found in 20!.

Thus:

Rewrite the factorials in multiplicative order.

In this scenario, the numbers of the factorial in the numerator and denominator CANNOT cancel. Simplify by multiplying out the factorials.

n! indicates a factorial, which means the products of numbers from 1 to n.

Write out the terms for the factorial.

Cancel out the common numbers in the numerator and denominator.

The answer is .

To evaluate the product of these factorials, we will need to expand the factorials.

Multiply all the values.

The answer is:

Expand the factorials in the numerator and denominator.

The terms 1, 2, 3, and 4 can be eliminated from the numerator and denominator. The simplification becomes:

Evaluate the factorials first.

Substitute the values back into the expression.

Simplify the numerator.

Evaluate each factorial inside the parentheses.

Zero factorial is equal to . Simplify by order of operations.

The answer is .

Simplify the following factorial:

To solve this, we need to understand factorials. Factorials are denoted by a number followed by the ! sign. What a factorial is, is a given number multiplied by each positive integer leading up to the given number. Stated differently...

So what we have above is really:

Now, because numbers 1 through 14 are in the numerator and the denominator, they will cancel and we will be left with:

If you were to write out the factorial problem above it would look like:

The only numbers that do not cancel out in the equation are the in the denominator. Therefore the answer is .

To simplify the expression involving factorials, we must remember what a factorial is:

For our factorial expression, we can write out some of the terms of the numerator's factorial and we will be able to simplify from there:

As you can see, n! remains on top and bottom after writing out the first three terms of the numerator's factorial. There is no need to expand any further once we have the same factorial on top and bottom. They cancel, and we get our final answer.

To simplify the expression, knowing that a factorial is, for example

will make it far easier.

Keeping this in mind, we can rewrite the given expression as

which simplifies to

after canceling like terms.

(One does not need to write out all terms of a factorial, rather as many in front that will leave something that cancels with another factorial are the most we usually need!)