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Algebra II : Factorials

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Example Question #211 : Mathematical Relationships And Basic Graphs

Try without a calculator.

True or false: $5! \times 5! = 25!$

Possible Answers:

False

True

Correct answer:

False

Explanation:

It is not necessary - and in fact, without a calculator, it is inconvenient - to calculate 25! to determine whether this is true or false.

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

$5! = 1 \times 2 \times 3 \times 4 \times 5 = 120$

$5! \times 5! =120 \times 120 = 14,400$

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

$6! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 = 720$

$7! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 = 5,040$

$8! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8= 40,320$

It can already be seen that and all higher factorials will be greater than $5! \times 5!$ , so it follows that the statement $5! \times 5! = 25!$ is false.

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Example Question #61 : Multiplying And Dividing Factorials

Simplify the expression.

$\frac{7!xy^2}{5!y}$

Possible Answers:

7!xy

42xy

$\frac{7!xy}{5!}$

7xy

Correct answer:

42xy

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.

$\frac{7!xy^2}{5!y}=\frac{7*6*5*4*3*2*1*x*y*y}{5*4*3*2*1*y}=7*6*x*y=42xy$

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Example Question #73 : Factorials

Try without a calculator.

Which expression is equal to $\frac{1,000!}{999!}$ ?

Possible Answers:

1,000

999!

999

None of these

Correct answer:

1,000

Explanation:

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

$1,000 ! = 1 \cdot 2 \cdot 3 \cdot 4 \cdot ...\cdot 999 \cdot 1,000$

and

$999 ! = 1 \cdot 2 \cdot 3 \cdot 4 \cdot ...\cdot 999$

Therefore,

$\frac{1,000!}{999!}$

is equal to

$\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot ...\cdot 999 \cdot 1,000}{1 \cdot 2 \cdot 3 \cdot 4 \cdot ...\cdot 999 }$

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

$\frac{ 1,000}{1 }=1,000$

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Example Question #1 : Understanding Factorials

What is the value of ?

Possible Answers:

Correct answer:

Explanation:

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

Thus, 4! = 4 * 3 * 2 * 1 = 24 .

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Example Question #2 : Other Factorials

What is the value of $\frac{10!}{7!}$ ?

Possible Answers:

None of the other answers are correct.

504

720

1.43

980

Correct answer:

720

Explanation:

Since the factorial has the property

$\frac{n!}{(n+1)!}$

we can write 10! as:

$10 \cdot 9 \cdot 8 \cdot (7!)$ .

Thus, our expression can be written as

$\frac{10\cdot9\cdot8\cdot7!}{7!}=10\cdot9\cdot8=720$

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Example Question #71 : 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, 4! = 4*3*2*1 which gives us .

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Example Question #1 : Other Factorials

Solve the Quadratic Equation.

4x^2 = 12x - 5

Possible Answers:

1, 1

2/5, 2

3/4, 1/2

{5/2, 1/2}

5/2, 2/5

Correct answer:

{5/2, 1/2}

Explanation:

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

$4x^{2} = 12x - 5$

-(12x - 5) -(12x - 5) = 0

$= 4x^{2} - 12x + 5 = 0$

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

= (2x - 5) (2x - 1)

2x - 5 = 0

+ 5 + 5

2x = 5

x = 5/2

2x - 1 = 0

+1 +1

2x = 1

x = 1/2

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Example Question #5 : Other Factorials

What is $\frac{10!}{7!}$ ?

Possible Answers:

720

810

4305

Correct answer:

720

Explanation:

Remember that factorial is defined as:

$n! = n \cdot (n-1)\cdot (n-2) \cdots \cdot (3)\cdot (2)\cdot (1)$

So using this definition, 10! = 10*9*8*7*6*5*4*3*2*1

And 7! = 7*6*5*4*3*2*1 .

So $\frac{10!}{7!} = \frac{10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}$

The 7, 6, 5, 4, 3, 2, and 1 all cancel from both the numerator and denominator. So we're left with just 10*9*8 on top, which has a value of 720 .

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Example Question #6 : Other Factorials

How many -permutations (without repetition) are there when taking numbers from the set of numbers $\begin{Bmatrix} 16, 11, 9, 7, 14, 12, 18, 23, 1 \end{Bmatrix}$ ?

Possible Answers:

504

3024

729

Correct answer:

504

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 9*8*7 gives us 504 .

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Example Question #7 : Other Factorials

What can n!(n+1) be expressed as?

Possible Answers:

n(n+1)

n+1

n(n+1)!

(n+1)!

Correct answer:

(n+1)!

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. n+1 is definitely greater than in this situation because factorials are always positive numbers. If we took the difference between n+1 and we would get . This means after , the next biggest value is n+1 . Since we have and the next value multiplied is n+1 , we can conclude that (n+1)! will work since it incorporates values multiplied and also n+1 .

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Algebra II : Factorials

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #211 : Mathematical Relationships And Basic Graphs

Example Question #61 : Multiplying And Dividing Factorials

Example Question #73 : Factorials

Example Question #1 : Understanding Factorials

Example Question #2 : Other Factorials

Example Question #71 : Factorials

Example Question #1 : Other Factorials

Example Question #5 : Other Factorials

Example Question #6 : Other Factorials

Example Question #7 : Other Factorials

All Algebra II Resources

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