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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #1 : Multiplying And Dividing Factorials

Which of the following is NOT the same as $\frac{6!\times5!}{4!}$ ?

Possible Answers:

$6\times5!$

$6\times5^{2}\times4!$

$\frac{6\times(5!)^2}{4!}$

$5\times6!$

$30\times5!$

Correct answer:

$6\times5!$

Explanation:

The cancels out all of except for the parts higher than 4, this leaves a 6 and a 5 left to multilpy

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Example Question #1 : Factorials

Simplify the following expression:

$\frac{7!(n+3)!}{6!(n+2)!}$

Possible Answers:

7!(n+3)

n+3

7n + 21

$\frac{n+3}{n+2}$

7n +3

Correct answer:

7n + 21

Explanation:

Recall that (n+3)! = (n+3)(n+2)(n+1)(n)... .

Likewise, $6! = 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1$ .

Thus, the expression $\frac{7!(n+3)!}{6!(n+2)!}$ can be simplified in two parts:

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

and

$\frac{(n+3)!}{(n+2)!} = \frac{(n+3)(n+2)(n+1...)}{(n+2)(n+1)...} = n+3$

The product of these two expressions is the final answer: 7n + 21

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Example Question #3 : Factorials

$\frac{6!}{2!4!} = ?$

Possible Answers:

$\frac{3}{4}$

Correct answer:

Explanation:

To simplify this, just write out each factorial:

$\frac{6!}{2!4!} = \frac{6*5*4*3*2*1}{(2*1)(4*3*2*1)} = \frac{6*5}{2*1} = 15$

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Example Question #1 : Multiplying And Dividing Factorials

Find the value of:

$\frac{7!}{3!}$

Possible Answers:

840

210

2.33

2520

Correct answer:

840

Explanation:

The factorial sign (!) just tells us to multiply that number by every integer that leads up to it. So, $\frac{7!}{3!}$ can also be written as:

$\frac{(7\times6\times5\times4\times3\times2\times1)}{(3\times2\times1)}$

To make this easier for ourselves, we can cancel out the numbers that appear on both the top and bottom:

$7\times6\times5\times4 = 840$

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Example Question #2 : Factorials

Which of the following is equivalent to $6!\times4!$ ?

Possible Answers:

$(6!)^{2}-4!$

24!

None of the other answers are correct.

$30\times(4!)^2$

10!

Correct answer:

$30\times(4!)^2$

Explanation:

This is a factorial question. The formula for factorials is $n! = n\times(n-1)\times(n-2)\times...\times1$ .

$6!=6\times5\times4! = 30\times 4!$

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Example Question #3 : Factorials

Divide 2015! by 2014!

Possible Answers:

2015

21,900,320

1,000,000

4,058,210

3,870,890

Correct answer:

2015

Explanation:

A factorial is a number which is the product of itself and all integers before it. For example $5!=5\cdot 4 \cdot 3 \cdot 2 \cdot 1$

In our case we are asked to divide 2015! by 2014! . To do this we will set up the following:

$\frac{2015!}{2014!}$

We know that 2015! can be rewritten as the product of itself and all integers before it or:

$2015!=2015 \cdot 2014!$

Substituting this equivalency in and simplifying the term, we get:

$\frac{2015 \cdot 2014!}{2014!}=2015$

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Example Question #1 : Factorials

$\frac{\left ( n+2\right )!}{n!}$

If is a postive integer, which of the following answer choices is a possible value for the expression.

Possible Answers:

104

Correct answer:

Explanation:

This expression of factorials reduces to (n+1)(n+2). Therefore, the solution must be a number that multiplies to 2 consecutive integers. Only 30 is a product of 2 consecutive integers. $5\ast6=30$

So n would have to be 4 in this problem.

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Example Question #1 : Multiplying And Dividing Factorials

Simplify:

$\frac{20!}{17!\times 2!}$

Possible Answers:

1.18

0.68

1.05

3420

6840

Correct answer:

3420

Explanation:

Remember what a factorial is, and first write out what the original equation means. A factorial is a number that you multiply by all whole numbers that come before it until you reach one.

You can simplify because all terms in the expression 17! are found in 20!.

Thus:

$\frac{20!}{17!\times 2!}=\frac{20\cdot19\cdot18\cdot17\cdot16\cdot15\cdot14\cdot13\cdot12\cdot11\cdot10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}{17\cdot16\cdot15\cdot14\cdot13\cdot12\cdot11\cdot10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1\cdot 2}$

$=\frac{20\cdot19\cdot18}{2}=10\cdot19\cdot18=3420$

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Example Question #10 : Multiplying And Dividing Factorials

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

Possible Answers:

$\frac{5}{4}$

$\frac{25}{7}$

$\frac{17}{10}$

Correct answer:

$\frac{25}{7}$

Explanation:

Rewrite the factorials in multiplicative order.

$\frac{4!+1}{3!+1} = \frac{(4\times 3\times2\times1)+1}{(3\times2\times1)+1}$

In this scenario, the numbers of the factorial in the numerator and denominator CANNOT cancel. Simplify by multiplying out the factorials.

$\frac{(4\times 3\times2\times1)+1}{(3\times2\times1)+1}= \frac{24+1}{6+1}=\frac{25}{7}$

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Example Question #11 : Factorials

Simplify the following expression:

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

Possible Answers:

$\frac{1296}{25}$

120

$\frac{12}{5}$

$\frac{10}{7}$

Correct answer:

Explanation:

n! indicates a factorial, which means the products of numbers from 1 to n.

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

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

$=6\cdot 4\cdot 3=72$

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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Multiplying And Dividing Factorials

Example Question #1 : Factorials

Example Question #3 : Factorials

Example Question #1 : Multiplying And Dividing Factorials

Example Question #2 : Factorials

Example Question #3 : Factorials

Example Question #1 : Factorials

Example Question #1 : Multiplying And Dividing Factorials

Example Question #10 : Multiplying And Dividing Factorials

Example Question #11 : Factorials

All Algebra II Resources

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