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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #2871 : Algebra Ii

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

Possible Answers:

120

Correct answer:

Explanation:

Simplify the inner parentheses and expand the factorials.

$(3!)(3+2)!(\frac{1}{4!}) = (3\times 2\times 1)(5)!(\frac{1}{4 \times 3\times 2\times 1})$

Cancel out common terms.

$(5\times 4\times 3\times 2\times 1)(\frac{1}{4}) = 5\times 3\times 2\times 1$

The answer is:

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Example Question #61 : Factorials

Multiply: $\frac{2!}{3!}\cdot \frac{4!}{6!}$

Possible Answers:

$\frac{1}{90}$

$\frac{2}{15}$

$\frac{1}{60}$

$\frac{1}{30}$

Correct answer:

$\frac{1}{90}$

Explanation:

In order to solve this, we will need to expand all the factorials.

$\frac{2!}{3!}\cdot \frac{4!}{6!} = \frac{2\times1}{3\times2\times1}\cdot \frac{4\times3\times2\times1}{6\times5\times4\times3\times2\times1 }$

Cancel all the common terms and write the remaining numbers.

$\frac{1}{3 }\cdot \frac{1}{30} = \frac{1}{90}$

The answer is: $\frac{1}{90}$

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Example Question #61 : Multiplying And Dividing Factorials

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

Possible Answers:

$\frac{18}{5}$

$\frac{36}{5}$

47900160

Correct answer:

$\frac{36}{5}$

Explanation:

In order to simplify the factorials, expand all the terms first.

$\frac{3(4!)}{2!+2!+3!} = \frac{3(4\times 3\times2\times 1)}{(2\times 1)+(2\times 1)+(3\times2\times 1)}$

Simplify the numerator and denominator.

$\frac{3(24)}{2+2+6} = \frac{72}{10}=\frac{36}{5}$

The answer is: $\frac{36}{5}$

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Example Question #2871 : Algebra Ii

Multiply: $0!\cdot (3+2!)\cdot 0!$

Possible Answers:

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

120

Correct answer:

Explanation:

Evaluate by expanding the factorial in the parentheses. The zero factorial is a special case which equals to one.

$0!\cdot (3+2!)\cdot 0! = 1\cdot (3+(2\times 1))\cdot 1 = 1\cdot (5) \cdot 1$

The answer is:

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Example Question #63 : Multiplying And Dividing Factorials

Divide: $\frac{3!}{5!}\div 3!$

Possible Answers:

$\frac{3}{40}$

$\frac{1}{120}$

$\frac{1}{40}$

$\frac{1}{60}$

$\frac{3}{10}$

Correct answer:

$\frac{1}{120}$

Explanation:

Evaluate by expanding the terms of the factorials.

$\frac{3!}{5!}\div 3! = \frac{3\times2\times1 }{5\times4\times3\times2\times1 }\div (3\times2\times1 )$

Cancel out the common terms in the first fraction.

$\frac{1 }{5\times4 }\div (3\times2\times1 )$

Change the division sign to a multiplication and take the reciprocal of the second quantity.

$\frac{1 }{5\times4 }\times \frac{1}{(3\times2\times1 )} =\frac{1}{120}$

The answer is: $\frac{1}{120}$

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Example Question #62 : Multiplying And Dividing Factorials

Solve: $\frac{(5+3)!}{2(2+2)!}$

Possible Answers:

420

1680

440

840

Correct answer:

840

Explanation:

Evaluate the terms in the parentheses first.

$\frac{(5+3)!}{2(2+2)!} =\frac{(8)!}{2(4)!}$

Expand the factorials.

$\frac{8\times7\times6\times5\times4\times3\times2\times1 }{2(4\times3\times2\times1 )}$

The fraction, after simplifying all the terms, becomes:

$\frac{8\times7\times6\times5}{2} = 4\times7\times6\times5 = 840$

The answer is: 840

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Example Question #64 : Multiplying And Dividing Factorials

Solve the factorials: $\frac{3!}{5!} \cdot \frac{(1+5)!}{(3!+1)}$

Possible Answers:

$\frac{121}{480}$

600

$\frac{3}{2}$

$\frac{121}{140}$

$\frac{36}{7}$

Correct answer:

$\frac{36}{7}$

Explanation:

Simplify the terms in parentheses first.

$\frac{3!}{5!} \cdot \frac{(1+5)!}{(3!+1)} = \frac{3!}{5!} \cdot \frac{6!}{(3!+1)}$

Evaluate each factorial by writing out the terms.

$\frac{3\times 2\times 1}{5\times 4\times 3\times 2\times 1} \cdot \frac{6\times 5\times 4\times 3\times 2\times 1}{([3\times 2\times 1]+1)}$

Simplify the parentheses.

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

Simplify all the common terms.

$\frac{6\times 3\times 2\times 1}{7} = \frac{36}{7}$

The answer is: $\frac{36}{7}$

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Example Question #64 : Multiplying And Dividing Factorials

Try without a calculator.

True or false: $5! \times 5! = 25!$

Possible Answers:

True

False

Correct answer:

False

Explanation:

It is not necessary - and in fact, without a calculator, it is inconvenient - to calculate 25! to determine whether this is true or false.

- or factorial - is defined to be the product of the integers from 1 to . Therefore,

$5! = 1 \times 2 \times 3 \times 4 \times 5 = 120$

$5! \times 5! =120 \times 120 = 14,400$

If we continue to look at factorials, we can see that

$6! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 = 720$

$7! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 = 5,040$

$8! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8= 40,320$

It can already be seen that and all higher factorials will be greater than $5! \times 5!$ , so it follows that the statement $5! \times 5! = 25!$ is false.

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Example Question #2872 : Algebra Ii

Simplify the expression.

$\frac{7!xy^2}{5!y}$

Possible Answers:

42xy

7xy

$\frac{7!xy}{5!}$

7!xy

Correct answer:

42xy

Explanation:

By expanding the factorials and the powers it is a lot easier to see what terms will cancel.

In this example everything in the denominator cancels leaving a 6,7,x, and y in the numerator.

$\frac{7!xy^2}{5!y}=\frac{7*6*5*4*3*2*1*x*y*y}{5*4*3*2*1*y}=7*6*x*y=42xy$

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Example Question #71 : Multiplying And Dividing Factorials

Try without a calculator.

Which expression is equal to $\frac{1,000!}{999!}$ ?

Possible Answers:

999!

1,000

999

None of these

Correct answer:

1,000

Explanation:

- or factorial - is defined to be the product of the integers from 1 to . Therefore,

$1,000 ! = 1 \cdot 2 \cdot 3 \cdot 4 \cdot ...\cdot 999 \cdot 1,000$

and

$999 ! = 1 \cdot 2 \cdot 3 \cdot 4 \cdot ...\cdot 999$

Therefore,

$\frac{1,000!}{999!}$

is equal to

$\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot ...\cdot 999 \cdot 1,000}{1 \cdot 2 \cdot 3 \cdot 4 \cdot ...\cdot 999 }$

All of the factors from 1 to 999 can be canceled out in both numerator and denominator, so the expression is equal to

$\frac{ 1,000}{1 }=1,000$

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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #2871 : Algebra Ii

Example Question #61 : Factorials

Example Question #61 : Multiplying And Dividing Factorials

Example Question #2871 : Algebra Ii

Example Question #63 : Multiplying And Dividing Factorials

Example Question #62 : Multiplying And Dividing Factorials

Example Question #64 : Multiplying And Dividing Factorials

Example Question #64 : Multiplying And Dividing Factorials

Example Question #2872 : Algebra Ii

Example Question #71 : Multiplying And Dividing Factorials

All Algebra II Resources

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