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Algebra II : Factorials

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Example Question #21 : Factorials

Simplify the following:

$\frac{(n+2)!}{n!}$

Possible Answers:

(n+2)(n+1)

(n-2)(n-1)

(n+2)(n+1)(n)

Correct answer:

(n+2)(n+1)

Explanation:

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

$4! = 4\cdot3\cdot2 \cdot 1 = 24$

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:

$\frac{(n+2)!}{n!}=\frac{(n+2)(n+1)(n)!}{n!}=(n+2)(n+1)$

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.

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Example Question #22 : Factorials

Simplify (leave as a product of binomials):

$\frac{(n+2)!}{n!}\cdot \frac{(n+3)!}{(n+1)!}$

Possible Answers:

(n+3)(n+2)^2(n+1)

(n+3)(n+2)(n+1)

(n+2)!(n+1)!(n)!

(n+1)!(n)!

Correct answer:

(n+3)(n+2)^2(n+1)

Explanation:

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

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

will make it far easier.

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

$\frac{(n+2)(n+1)(n!)}{n!}\cdot \frac{(n+3)(n+2)(n+1)!}{(n+1)!}$

which simplifies to

(n+3)(n+2)^2(n+1)

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!)

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Example Question #23 : Factorials

Simplify:

$\frac{(n-3)!}{3!}\cdot \frac{5!}{(n-4)!}$

Possible Answers:

$\frac{n-4}{20}$

20(n-3)

$\frac{20}{n-4}$

$\frac{20(n-3)}{n-4}$

Correct answer:

$\frac{20}{n-4}$

Explanation:

To simplify, remember that a factorial simply means we take the given term and subtract one unit from it successively until we reach 1, and multiply all the terms together:

$n!=n(n-1)(n-2)\cdot...\cdot 3\cdot2\cdot1$

Rewriting our given expression, we get:

$\frac{(n-3)!}{3!} \cdot \frac{5 \cdot4\cdot3!}{(n-4)(n-3)!}$

Notice how we only expanded the factorials as much as we needed to in order to cancel things:

$\frac{5\cdot 4}{n-4}=\frac{20}{n-4}$

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Example Question #161 : Mathematical Relationships And Basic Graphs

Simplify the expression $\frac{8!}{5!}$

Possible Answers:

120

336

Correct answer:

336

Explanation:

$\frac{8!}{5!}=\frac{8*7*6*5*4*3*2*1}{5*4*3*2*1}$

The 5,4,3,2,1 cancel out leaving 8*7*6 which is 336

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Example Question #25 : Factorials

Multiply the factorials: $4!\times 3!$

Possible Answers:

144

$\textup{The answer is not given.}$

169

$4.79\times 10^{8}$

Correct answer:

144

Explanation:

Rewrite the factorials.

$4! = 4\times3\times2\times1 = 24$

$3!=3\times2\times1 = 6$

Multiply these two numbers together.

$24\times 6 = 144$

The answer is: 144

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Example Question #26 : Factorials

Divide the factorials: $\frac{6!}{3!}$

Possible Answers:

120

360

720

240

Correct answer:

120

Explanation:

Expand both factorials in the numerator and denominator.

$\frac{6!}{3!} = \frac{6 \times 5\times4\times3\times2\times1}{3\times2\times1}$

Cancel the common terms on the numerator and denominator.

The numerator becomes:

$6\times 5 \times 4 =120$

The answer is: 120

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Example Question #27 : Factorials

Simplify the factorials: $\frac{3!(3!+2!)^2}{5! }$

Possible Answers:

$\frac{32}{5}$

$\frac{16}{5}$

720

$\frac{120}{7}$

Correct answer:

$\frac{16}{5}$

Explanation:

Do not add the factorials!

$3!+2! \neq 5!$

We will need to simplify and rewrite the terms of the factorials.

$\frac{3!(3!+2!)^2}{5!} = \frac{(3\times 2\times 1)[(3\times 2\times 1)+(2\times 1)]^2}{5\times 4\times3\times 2\times 1}$

The common terms in the numerator and denominator can be simplified.

$\frac{[(3\times 2\times 1)+(2\times 1)]^2}{5\times 4}$

Simplify the numerator and denominator.

$\frac{[6+2]^2}{20} = \frac{8^2}{20} = \frac{64}{20} =\frac{4\times 4\times 4}{5\times 4}$

Simplify the fraction by cancelling the common terms.

The answer is: $\frac{16}{5}$

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Example Question #28 : Factorials

Solve: $3!(\frac{(4+3)!}{7(5!)})$

Possible Answers:

$\frac{1}{20}$

$\frac{1}{40}$

$\frac{1}{14}$

Correct answer:

Explanation:

Rewrite all the factorial terms in expanded form. To expand a factorial, multiply all the integers in decreasing order until it reaches to one.

$3!(\frac{(4+3)!}{7(5!)}) = (3\times 2\times 1)(\frac{7!}{7(5!)}) = 6(\frac{7!}{7(5!)})$

Do not distribute the integer through the number before the factorial, and do not cancel the sevens in the numerator and denominator.

$7(5!) \neq 35!$

$\frac{7!}{7} \neq 1!$

Rewrite the numerator in terms so that we can simplify the fraction.

$6(\frac{7!}{7(5!)}) = 6(\frac{7\times 6\times5\times4\times3\times2\times1}{7(5\times4\times3\times2\times1)})$

Notice all the constants that we can cancel. This fraction will reduce to"

$6\times 6 = 36$

The answer is:

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Example Question #21 : Multiplying And Dividing Factorials

Multiply the factorials: $3!\times (2+3)!\times 0!$

Possible Answers:

1296

120

720

Correct answer:

720

Explanation:

Expand the factorials.

$3!\times (2+3)!\times 0! = (3\times 2\times 1)\times 5! \times 0!$

$=(3\times 2\times 1)\times(5\times 4\times 3\times 2\times 1)\times 0!$

Zero factorial is a special case. It is equal to one.

0!=1

The expression becomes:

$=(3\times 2\times 1)\times(5\times 4\times 3\times 2\times 1)$

Multiply all the numbers.

$=6\times 120= 720$

The answer is: 720

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Example Question #30 : Factorials

Simplify the factorials: $\frac{2!(3+3)!}{ 3!\times 6}$

Possible Answers:

$\frac{2}{3}$

$\frac{1}{3}$

Correct answer:

Explanation:

Simplify the factorials in the numerator first.

$2!(3+3)! = 2!\cdot 6! =(2\times 1)\cdot (6\times5\times4\times3\times2\times1)$

Simplify the denominator.

$3!\times 6 = (3\times 2\times 1) \times 6$

Eliminate common terms in the numerator and denominator.

The numerator becomes: $2 \times 1\times5\times4 = 40$

The answer is:

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Algebra II : Factorials

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #21 : Factorials

Example Question #22 : Factorials

Example Question #23 : Factorials

Example Question #161 : Mathematical Relationships And Basic Graphs

Example Question #25 : Factorials

Example Question #26 : Factorials

Example Question #27 : Factorials

Example Question #28 : Factorials

Example Question #21 : Multiplying And Dividing Factorials

Example Question #30 : Factorials

All Algebra II Resources

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