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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

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10 Diagnostic Tests 630 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #22 : Multiplying And Dividing Factorials

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

Possible Answers:

720

360

240

120

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 #22 : Factorials

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

Possible Answers:

720

120

1296

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 #23 : Multiplying And Dividing Factorials

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

Possible Answers:

29520

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

24678720

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 #171 : Mathematical Relationships And Basic Graphs

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

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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Example Question #33 : Factorials

$\frac{7!}{4!}\cdot \frac{5!}{2!}$

Possible Answers:

210

24628

12600

1020

120

Correct answer:

12600

Explanation:

Similar factors that are in both the numerator and denominator of a fraction cancel out. It helps to write it in expanded form.

$\frac{7!}{4!}\cdot \frac{5!}{2!}$

$\frac{7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{4\cdot 3\cdot 2\cdot 1}\cdot \frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{2\cdot 1}$

$7\cdot 6\cdot 5 \cdot 5\cdot 4\cdot 3$

12600

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Example Question #34 : Factorials

$\frac{6!\cdot 4!\cdot 2!}{10!}$

Possible Answers:

$\frac{3}{110}$

728

$\frac{1}{105}$

1020

120

Correct answer:

$\frac{1}{105}$

Explanation:

Common factors on the numerator and denominator of a fraction will cancel. It helps to write problems in expanded form.

$\frac{6!\cdot 4!\cdot 2!}{10!}$

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

$\frac{4\cdot 3\cdot 2\cdot 1\cdot 2\cdot 1}{10\cdot 9\cdot 8\cdot 7}$

$\frac{48}{5040}$

Then, reduce the fraction to lowest terms. Both numerator and denominator divide by 48.

$\frac{1}{105}$

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Example Question #35 : Factorials

$\frac{4!}{3!\cdot 2!}$

Possible Answers:

$\frac{1}{2}$

$\frac{2}{3}$

$\frac{1}{12}$

Correct answer:

Explanation:

Common factors in the numerator and denominator will cancel. It helps to write factorials in expanded form.

$\frac{4!}{3!\cdot 2!}$

$\frac{4\cdot 3\cdot 2\cdot 1}{3\cdot 2\cdot 1\cdot 2\cdot 1}$

$\frac{4}{2\cdot 1}$

$\frac{4}{2}$

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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #22 : Multiplying And Dividing Factorials

Example Question #27 : Factorials

Example Question #28 : Factorials

Example Question #22 : Factorials

Example Question #30 : Factorials

Example Question #23 : Multiplying And Dividing Factorials

Example Question #171 : Mathematical Relationships And Basic Graphs

Example Question #33 : Factorials

Example Question #34 : Factorials

Example Question #35 : Factorials

All Algebra II Resources

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