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Algebra II : Multiplying and Dividing Factorials

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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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Example Question #31 : Factorials

Simplify the factorials: $(7-4)!\cdot(7!-5!)$

Possible Answers:

29520

24678720

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

Correct answer:

29520

Explanation:

Evaluate the first term. Write out the terms of the factorial.

$(7-4)! = 3! = 3\times 2 \times 1 = 6$

Evaluate the second term.

(7!-5!)

$7! = 7\times6\times5\times4\times3\times2\times1 = 5040$

$5!=5\times4\times3\times2\times1 = 120$

(7!-5!) = 5040-120 = 4920

Combine the simplified terms.

$(7-4)!\cdot(7!-5!) = 6\cdot 4920$

The answer is: 29520

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Example Question #32 : Factorials

$\frac{15!}{13!}$

Possible Answers:

210

6227020800

375

Correct answer:

210

Explanation:

When you have factorials in the numerator and denominator of a fraction, you can cancel out the common factors between them. It can help to write the problems in expanded form.

$\frac{15!}{13!}$

$\frac{15\cdot 14\cdot 13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}$

The entire denominator 13 will be cancelled out leaving only

$15\cdot 14$

210

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Algebra II : Multiplying and Dividing Factorials

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

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

Example Question #31 : Factorials

Example Question #32 : Factorials

All Algebra II Resources

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