# Algebra II : Factorials

## Example Questions

### Example Question #71 : Factorials

Try without a calculator.

True or false:

False

True

False

Explanation:

It is not necessary - and in fact, without a calculator, it is inconvenient - to calculate  to determine whether this is true or false.

- or  factorial - is defined to be the product of the integers from 1 to . Therefore,

If we continue to look at factorials, we can see that

It can already be seen that  and all higher factorials will be greater than , so it follows that the statement  is false.

### Example Question #72 : Factorials

Simplify the expression.

Explanation:

By expanding the factorials and the powers it is a lot easier to see what terms will cancel.

In this example everything in the denominator cancels leaving a 6,7,x, and y in the numerator.

### Example Question #73 : Factorials

Try without a calculator.

Which expression is equal to  ?

None of these

Explanation:

- or  factorial - is defined to be the product of the integers from 1 to . Therefore,

and

Therefore,

is equal to

All of the factors from 1 to 999 can be canceled out in both numerator and denominator, so the expression is equal to

### Example Question #1 : Understanding Factorials

What is the value of  ?

6

12

4

24

10

24

Explanation:

! is the symbol for factorial, which means the product of the whole numbers less than the given number.

Thus,

### Example Question #1 : Other Factorials

What is the value of ?

None of the other answers are correct.

Explanation:

Since the factorial has the property

we can write  as:

.

Thus, our expression can be written as

### Example Question #2 : Other Factorials

What is the value of  ?

Explanation:

A factorial represents the product of all natural numbers less than a given number. Thus,  which gives us

### Example Question #3 : Other Factorials

Explanation:

First, move all terms to the left by subtracting the quantity on the right.

From here, factor the quadractic into two binomials and set each equal to zero.

### Example Question #4 : Other Factorials

What is ?

Explanation:

Remember that factorial is defined as:

So using this definition,

And

So

The 7, 6, 5, 4, 3, 2, and 1 all cancel from both the numerator and denominator. So we're left with just  on top, which has a value of .

### Example Question #5 : Other Factorials

How many -permutations (without repetition) are there when taking numbers from the set of numbers ?

Explanation:

The elements of the set don't matter. Only the size of the set matters when determining permutations.

Our set contains 9 integers, so for the first number in our permutation, we have 9 choices.

After picking that number, because we're not allowed repetition, our second number is from 8 choices.

Our final number is from 7 choices.

Multiplying  gives us .

### Example Question #6 : Other Factorials

What can   be expressed as?