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Algebra II : Mathematical Relationships and Basic Graphs

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Example Questions

Example Question #191 : Mathematical Relationships And Basic Graphs

Multiply: 4(2!)(3!)(2+3!)

Possible Answers:

720

345600

384

5760

1935360

Correct answer:

384

Explanation:

Simplify all the factorials first.

$2!= 2\times 1 = 2$

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

Substitute each term.

4(2!)(3!)(2+3!) = 4(2)(6)(2+6)

Simplify by order of operations.

48(8) = 384

The answer is: 384

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Example Question #41 : Factorials

Divide the factorials: $\frac{4(4!)}{3(3!)^2}$

Possible Answers:

$\frac{32}{9}$

$\frac{8}{9}$

$\frac{1}{11340}$

$1.92\times 10^7$

$\frac{1430}{9}$

Correct answer:

$\frac{8}{9}$

Explanation:

Expand the factorials first. Do not distribute the integers inside the parentheses or this will change the value of the factorial.

$\frac{4(4!)}{3(3!)^2}=\frac{4(4\times 3\times 2\times 1)}{3(3\times 2\times 1)^2}=\frac{4(4\times 3\times 2\times 1)}{3(3\times 2\times 1)(3\times 2\times 1)}$

Cross out the common terms in the numerator and denominator.

$\frac{4(4)}{3(3\times 2\times 1)} =\frac{16}{18} = \frac{8}{9}$

The answer is: $\frac{8}{9}$

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Example Question #48 : Factorials

Multiply the factorials: (0!)(2!)(3!)!

Possible Answers:

360

1440

720

Correct answer:

1440

Explanation:

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

Expand the other factorials.

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

Simplify this expression.

$(1)(2)(6)! = 2(6\times 5\times4\times3\times2\times1) = 2(720) = 1440$

The answer is: 1440

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Example Question #49 : Factorials

Divide the factorials: $\frac{2!}{(0!+0!+0!)!}$

Possible Answers:

$\frac{2}{3}$

$\frac{1}{3}$

$\frac{1}{6}$

$\textup{Undefined}$

Correct answer:

$\frac{1}{3}$

Explanation:

The value of zero factorial is equal to one. This is the special case.

Simplify the expression.

$\frac{2!}{(0!+0!+0!)!} = \frac{2\times 1}{(1+1+1)!} = \frac{2}{3!}$

Write out the terms in the denominator.

$\frac{2}{3!}=\frac{2}{3\times 2\times 1}= \frac{1}{3}$

The answer is: $\frac{1}{3}$

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Example Question #50 : Factorials

Divide the factorials: $\frac{5!}{7!}\div\frac{5}{7}$

Possible Answers:

$\frac{1}{6}$

$\frac{5}{6}$

$\frac{1}{30}$

Correct answer:

$\frac{1}{30}$

Explanation:

Simplify the factorials first. Expand the terms.

$\frac{5!}{7!}\div\frac{5}{7} =\frac{5\times 4\times 3\times 2\times 1 }{7\times 6\times 5\times 4\times 3\times 2\times 1}\div\frac{5}{7}$

Cancel the common terms.

$\frac{1}{42}\div\frac{5}{7}$

Change the sign to multiplication and take the reciprocal of the second fraction.

$\frac{1}{42}\div\frac{5}{7} = \frac{1}{42}\times \frac{7}{5} =\frac{1}{6\times 5} =\frac{1}{30}$

The answer is: $\frac{1}{30}$

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Example Question #51 : Factorials

$\frac{30!\cdot 2!}{30!}$

Possible Answers:

$\frac{1}{4}$

$\frac{2}{29}$

$\frac{30}{29}$

Correct answer:

Explanation:

Simplify the factorials in each fraction by canceling common factors in the numerator and denominator. It would be helpful to write this in expanded form but the factorials are too large for this to be feasible.

$\frac{30!\cdot 2!}{30!}$

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

$\frac{2}{1}$

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Example Question #52 : Factorials

$\frac{6!}{4!}\div \frac{3!}{5!}$

Possible Answers:

600

$\frac{2}{3}$

$\frac{3}{2}$

$\frac{1}{10}$

Correct answer:

600

Explanation:

First, simplify the factorials in each fraction by canceling common factors in the numerator and denominator. It can help to write it in expanded form.

$\frac{6!}{4!}\div \frac{3!}{5!}$

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

$\frac{6\cdot 5}{1}\div \frac{1}{5\cdot 4}$

$\frac{30}{1}\div \frac{1}{20}$

You can now turn this into a multiplication problem and flip the second fraction

$\frac{30}{1}\cdot \frac{20}{1}$

600

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Example Question #53 : Factorials

$5!\cdot \frac{4!}{2!}\cdot \frac{2!}{6!}$

Possible Answers:

$\frac{1}{6}$

$\frac{1}{24}$

Correct answer:

Explanation:

First, simplify the factorials in each fraction by canceling common factors in the numerator and denominator. It can help to write it in expanded form.

$5!\cdot \frac{4!}{2!}\cdot \frac{2!}{6!}$

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

It is easiest to combine these numbers into one fraction. The 5! will go into the numerator

$\frac{5\cdot 4\cdot 3\cdot 2\cdot 1\cdot 4\cdot 3\cdot 2\cdot 1\cdot 2\cdot 1}{2\cdot 1\cdot 6\cdot5\cdot 4\cdot 3\cdot 2\cdot 1 }$

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

$\frac{24}{6}$

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Example Question #51 : Multiplying And Dividing Factorials

$\frac{3!}{5!}\cdot \frac{6!}{7!}\div \frac{8!}{6!}$

Possible Answers:

$\frac{1}{7840}$

$\frac{19}{5}$

$\frac{5}{19}$

$\frac{1}{5600}$

Correct answer:

$\frac{1}{7840}$

Explanation:

Simplify the factorials in each fraction by canceling common factors in the numerator and denominator. It can help to write it in expanded form.

$\frac{3!}{5!}\cdot \frac{6!}{7!}\div \frac{8!}{6!}$

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

$\frac{1}{5\cdot 4}\cdot \frac{1}{7}\div \frac{8\cdot 7}{1}$

$\frac{1}{20}\cdot \frac{1}{7}\div \frac{56}{1}$

$\frac{1}{140}\div \frac{56}{1}$

$\frac{1}{140}\cdot \frac{1}{56}$

$\frac{1}{7840}$

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Example Question #55 : Factorials

$\frac{20!\cdot 10!}{25!\cdot 5!}$

Possible Answers:

13080

$\frac{9}{13056}$

4080

$\frac{6}{1265}$

$\frac{1}{280795}$

Correct answer:

$\frac{6}{1265}$

Explanation:

Simplify the factorials the fraction by canceling common factors in the numerator and denominator. Normally, it would help to write out the factorials in expanded form, but since these are larger factorials that is not feasible. The best approach to cancelling these is thinking that 20! will be completely gone by taking it out of 25! and 5! will be completely gone by taking it out of 10!

$\frac{20!\cdot 10!}{25!\cdot 5!}$

$\frac{10\cdot 9\cdot 8\cdot 7\cdot 6}{25\cdot 24\cdot 23\cdot 22\cdot 21}$

$\frac{30240}{6375600}$

Then reduce this to lowest terms.

$\frac{3024}{637560}=\frac{1512}{318780}=\frac{756}{159390}=\frac{378}{79695}=\frac{54}{11385}=\frac{6}{1265}$

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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 #191 : Mathematical Relationships And Basic Graphs

Example Question #41 : Factorials

Example Question #48 : Factorials

Example Question #49 : Factorials

Example Question #50 : Factorials

Example Question #51 : Factorials

Example Question #52 : Factorials

Example Question #53 : Factorials

Example Question #51 : Multiplying And Dividing Factorials

Example Question #55 : Factorials

All Algebra II Resources

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