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Algebra II : Multiplying and Dividing Factorials

Study concepts, example questions & explanations for Algebra II

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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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Example Question #36 : Factorials

Divide the factorials: $\frac{8!}{(2+4)!}$

Possible Answers:

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

Correct answer:

Explanation:

Simplify the denominator in the parentheses first.

$\frac{8!}{(2+4)!} = \frac{8!}{6!}$

Rewrite the factorials.

$\frac{8!}{6!} = \frac{8\times7\times6\times5\times4\times3\times2\times1 }{6\times5\times4\times3\times2\times1}$

Cancel all the common terms in the numerator and denominator.

The numerator becomes: $8\times 7 =56$

The answer is:

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Example Question #37 : Factorials

Multiply the following factorials: $(1+2)! \cdot 3!$

Possible Answers:

Correct answer:

Explanation:

In order to simplify the factorials, first simplify the parentheses.

$(1+2)! \cdot 3!= 3! \cdot 3!$

Simplify the factorials.

$(3\times 2 \times 1)\cdot(3\times 2 \times 1) = (6) \cdot (6) =36$

The answer is:

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Example Question #38 : Factorials

Multiply the factorials: $\frac{(1+0!)!}{0!+3!}$

Possible Answers:

$\frac{1}{3}$

$\frac{1}{12}$

$\frac{2}{7}$

$\frac{1}{6}$

$\frac{1}{4}$

Correct answer:

$\frac{2}{7}$

Explanation:

The value of zero factorial is one.

0!=1

Replace this value.

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

Expand the factorials.

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

The answer is: $\frac{2}{7}$

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Example Question #31 : Factorials

Divide the factorials: $\frac{(3+5)!}{6!}$

Possible Answers:

$\frac{4}{3}$

$\frac{7}{40}$

Correct answer:

Explanation:

Simplify the numerator of the fraction.

$\frac{(3+5)!}{6!} = \frac{8!}{6!}$

Do not reduce the fraction and cancel out the factorials. Instead, expand both.

$\frac{8!}{6!} = \frac{8\times7\times6\times5\times4\times3\times2\times1}{6\times5\times4\times3\times2\times1}$

Notice that the similar terms on the numerator and denominator will cancel.

The remaining terms are: $8\times 7= 56$

The answer is:

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Example Question #40 : Factorials

Divide the factorials: $\frac{4!(4-2)!}{(1+2)!}$

Possible Answers:

120

6720

2240

Correct answer:

Explanation:

Simplify every term inside the parentheses.

$\frac{4!(4-2)!}{(1+2)!} = \frac{4!(2)!}{(3)!}$

Expand each factorial.

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

Cancel out the common terms on the numerator and denominator and rewrite what remains of the numerator.

$4\times 2\times 1=8$

The answer is:

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Example Question #41 : Factorials

Divide the factorials. $\frac{3! \cdot5!}{4!\cdot6!}$

Possible Answers:

$\frac{1}{72}$

$\frac{5}{8}$

$\frac{1}{48}$

$\frac{1}{24}$

$\frac{1}{36}$

Correct answer:

$\frac{1}{24}$

Explanation:

Expand each factorial.

$\frac{3! \cdot5!}{4!\cdot6!}=\frac{(3\times2\times1) (5\times4 \times3\times2\times1)}{ (4 \times3\times2\times1)(6\times 5\times4 \times3\times2\times1)}$

Cancel out the common terms on the numerator and denominator.

The remain terms are:

$\frac{1}{4(6)} = \frac{1}{24}$

The answer is: $\frac{1}{24}$

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Example Question #42 : Factorials

Divide the fractions: $\frac{3}{8}\div\frac{9}{34}$

Possible Answers:

$\frac{12}{17}$

$\frac{18}{23}$

$\frac{272}{27}$

$\frac{17}{12}$

$\frac{23}{18}$

Correct answer:

$\frac{17}{12}$

Explanation:

In order to divide these fractions, we will need to take the reciprocal of the second term and change the sign to a multiplication.

$\frac{3}{8}\times \frac{34}{9}$

Write each non-prime term in factors.

$\frac{3}{2\times 4}\times \frac{2\times 17}{3 \times 3}$

Reduce the terms by cancelling the common factors in the numerator and denominator.

$\frac{17}{4\times 3} = \frac{17}{12}$

The answer is: $\frac{17}{12}$

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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 #33 : Factorials

Example Question #34 : Factorials

Example Question #35 : Factorials

Example Question #36 : Factorials

Example Question #37 : Factorials

Example Question #38 : Factorials

Example Question #31 : Factorials

Example Question #40 : Factorials

Example Question #41 : Factorials

Example Question #42 : Factorials

All Algebra II Resources

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