# Algebra II : Factorials

## Example Questions

### Example Question #7 : Other Factorials

What is ?

Explanation:

A factorial means you are multiplying the number, with one less than previous number and you keep multiplying until you reach .

So,  is .

Multiplying that out, we get

### Example Question #8 : Other Factorials

What is ?

Explanation:

A factorial means you are multiplying the number, with one less than previous number and you keep multiplying until you reach

So,  is .

Multiplying that out, we get

### Example Question #9 : Other Factorials

What is ?

Explanation:

A factorial means you are multiplying the number, with one less than previous number and you keep multiplying until you reach

So,  is . Multiplying that out, we get .

We also have . That is  or .

Since we are multiplying the factorial, we multiply  and  to get

### Example Question #81 : Factorials

What is ?

Explanation:

A factorial means you are multiplying the number, with one less than previous number and you keep multiplying until you reach

So,  is . We also have . That is .

Since we are dividing the factorial, we can cancel out some terms.

Both the numerator and denominator have , and we can cancel those out. We are left with  or

### Example Question #82 : Factorials

What is ?

Explanation:

A factorial means you are multiplying the number, with one less than previous number and you keep multiplying until you reach

So,  is . We are dividing by  With careful inspection,  can be broken down to .

If we cancel that out with the ones in the numerator, we have  or

### Example Question #83 : Factorials

What is ?

Explanation:

A factorial means you are multiplying the number, with one less than previous number and you keep multiplying until you reach . Since we are dealing with variables, let's analyze each the numerator and the denominator.  is definitely greater than  in this situation because factorials are always positive numbers. If we took the difference between  and  we would get . This means that if we were to expand both the numerator and denominator, we cancel everything out except the extra term in the denominator which is .

### Example Question #84 : Factorials

Simplify.

Explanation:

A factorial means you are multiplying the number, with one less than previous number and you keep multiplying until you reach . Since we are dealing with variables, let's analyze each the numerator and the denominator.  is definitely greater than  in this situation because factorials are always positive numbers. If we took the difference between  and  we would get . This means that if we were to expand both the numerator and denominator, we cancel everything out except the  extra terms in the numerator which are . So final answer is .

If you need convincing, let . So we have .

You can cancel out the   from top and bottom to get . We need to express them into expressions.

Since  was , to get , you need to add  to  or

To get , you add  to  or

To get  you subtract  from  or .

You still get the same answer of

### Example Question #85 : Factorials

Simplify:

Explanation:

Write out the terms of each factorial in expanded form.  The only exception is the zero factorial, which is equal to one.

Simplify the terms.  The values in the numerator and denominator cannot cancel out!

### Example Question #86 : Factorials

What is the value of ?

Explanation:

Step 1: We must define factorial. Factorial is defined as , where

Step 2: Evaluate

Compute: