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Algebra II : Factorials

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #31 : Multiplying And Dividing Factorials

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

Possible Answers:

$\frac{1}{24}$

$\frac{1}{36}$

$\frac{5}{8}$

$\frac{1}{48}$

$\frac{1}{72}$

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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Example Question #43 : Factorials

Multiply the factorials: $0!\cdot 2! \cdot 4!$

Possible Answers:

720

Correct answer:

Explanation:

Identify the values of each factorial. Zero factorial is a special case.

0!=1

$2! = 2\times 1 = 2$

$4! =4\times 3 \times 2\times 1=24$

Multiply all three numbers.

$0!\cdot 2! \cdot 4!=1\cdot2\cdot24 = 48$

The answer is:

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Example Question #44 : Factorials

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

Possible Answers:

$\frac{1}{2}$

$1.186\times 10^{13}$

$\frac{3}{4}$

$\frac{5}{3}$

$\frac{5}{6}$

Correct answer:

$\frac{5}{6}$

Explanation:

Simplify the numerator. Do not distribute the five through the terms inside the parentheses, or it will change the value of the factorial!

$5(3+2)! = 5(5)! = 5[5\cdot 4\cdot 3\cdot 2\cdot 1]$

Simplify the denominator.

$5!\cdot 3! = [5\cdot 4\cdot 3\cdot 2\cdot 1]\cdot [3\cdot 2\cdot 1]$

Divide the numerator with the denominator.

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

The answer is: $\frac{5}{6}$

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Example Question #45 : Factorials

Simplify: $\frac{4(1+3)!}{2(3)!}$

Possible Answers:

$2.91\times 10^{10}$

$1.74\times 10^{11}$

6720

Correct answer:

Explanation:

Simplify the terms in side the parentheses first. Do NOT distribute the four through the binomial or this will change the value of the factorial!

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

Notice the there are common terms in the numerator and denominator. Cancel out the terms. The remaining parts are:

$\frac{4\cdot 4}{2} = 8$

The answer is:

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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 #41 : Factorials

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

Possible Answers:

$\frac{2}{3}$

$\frac{1}{3}$

$\textup{Undefined}$

$\frac{1}{6}$

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 #42 : Factorials

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

Possible Answers:

$\frac{1}{30}$

$\frac{1}{6}$

$\frac{5}{6}$

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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Algebra II : Factorials

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #31 : Multiplying And Dividing Factorials

Example Question #42 : Factorials

Example Question #43 : Factorials

Example Question #44 : Factorials

Example Question #45 : Factorials

Example Question #191 : Mathematical Relationships And Basic Graphs

Example Question #41 : Factorials

Example Question #48 : Factorials

Example Question #41 : Factorials

Example Question #42 : Factorials

All Algebra II Resources

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