Algebra II : Factorials

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #71 : Factorials

Try without a calculator.

True or false: 

Possible Answers:

False

True

Correct answer:

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. 

Possible Answers:

Correct answer:

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  ?

Possible Answers:

None of these

Correct answer:

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  ? 

Possible Answers:

6

12

4

24

10

Correct answer:

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 ?

Possible Answers:

None of the other answers are correct.

Correct answer:

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  ? 

Possible Answers:

Correct answer:

Explanation:

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

Example Question #3 : Other Factorials

Solve the Quadratic Equation.

Possible Answers:

Correct answer:

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 ?

Possible Answers:

Correct answer:

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 ?

Possible Answers:

Correct answer:

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?

Possible Answers:

Correct answer:

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 them.  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 after , the next biggest value is . Since we have  and the next value multiplied is , we can conclude that  will work since it incorporates  values multiplied and also .

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