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Algebra II : Factorials

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Example Question #9 : Multiplying And Dividing Factorials

Simplify:

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

Possible Answers:

1.05

1.18

6840

0.68

3420

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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Example Question #12 : Factorials

Divide: $\frac{4!}{2!}$

Possible Answers:

Correct answer:

Explanation:

Write out the terms for the factorial.

$\frac{4!}{2!} = \frac{4\times3\times2\times1}{2\times1}$

Cancel out the common numbers in the numerator and denominator.

$\frac{4\times3\times2\times1}{2\times1} = 4\times 3 = 12$

The answer is .

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Example Question #13 : Factorials

Multiply: $2! \cdot 4!$

Possible Answers:

720

40320

Correct answer:

Explanation:

To evaluate the product of these factorials, we will need to expand the factorials.

$2! \cdot 4! = (2\times 1) \cdot (4\times 3 \times 2\times 1)$

Multiply all the values.

$(2\times 1) \cdot (4\times 3 \times 2\times 1) = 2(24) =48$

The answer is:

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Example Question #14 : Factorials

Divide: $\frac{4!}{7!}$

Possible Answers:

$\frac{256}{823543}$

$\frac{4}{7}$

$\frac{2}{7}$

$\frac{1}{210}$

$\frac{64}{117649}$

Correct answer:

$\frac{1}{210}$

Explanation:

Expand the factorials in the numerator and denominator.

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

The terms 1, 2, 3, and 4 can be eliminated from the numerator and denominator. The simplification becomes:

$\frac{4!}{7!}=\frac{1}{7\times 6 \times 5} = \frac{1}{210}$

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Example Question #15 : Factorials

Solve: $\frac{2-(2-3!)}{0!}$

Possible Answers:

$\textup{Undefined}$

Correct answer:

Explanation:

Evaluate the factorials first.

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

0! = 1

Substitute the values back into the expression.

$\frac{2-(2-3!)}{0!} = \frac{2-(2-6)}{1}$

Simplify the numerator.

2-(2-6) = 2-2+6 = 6

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Example Question #16 : Factorials

Solve the factorial: $(\frac{0!-1}{0!+1})!$

Possible Answers:

$\textup{Undefined}$

Correct answer:

Explanation:

Evaluate each factorial inside the parentheses.

Zero factorial is equal to . Simplify by order of operations.

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

The answer is .

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Example Question #17 : Factorials

Simplify the following factorial:

$\frac{16!}{14!}$

Possible Answers:

365

240

Correct answer:

240

Explanation:

Simplify the following factorial:

$\frac{16!}{14!}$

To solve this, we need to understand factorials. Factorials are denoted by a number followed by the ! sign. What a factorial is, is a given number multiplied by each positive integer leading up to the given number. Stated differently...

5!=1*2*3*4*5

So what we have above is really:

$\frac{16!}{14!}=\frac{1*2*3*4*5*6*7*8*9*10*11*12*13*14*15*16}{1*2*3*4*5*6*7*8*9*10*11*12*13*14}$

Now, because numbers 1 through 14 are in the numerator and the denominator, they will cancel and we will be left with:

$\frac{16!}{14!}=\frac{15*16}{1}=240$

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Example Question #18 : Factorials

Find the value of $\frac{12!}{14!}$

Possible Answers:

$\frac{1}{14}$

$\frac{1}{13}$

$\frac{1}{2}$

$\frac{1}{182}$

182

Correct answer:

$\frac{1}{182}$

Explanation:

If you were to write out the factorial problem above it would look like:

$\frac{12\times 11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1}{14\times13\times12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1}$

The only numbers that do not cancel out in the equation are the $14\times13$ in the denominator. Therefore the answer is $\frac{1}{182}$ .

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Algebra II : Factorials

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #9 : Multiplying And Dividing Factorials

Example Question #10 : Multiplying And Dividing Factorials

Example Question #11 : Factorials

Example Question #12 : Factorials

Example Question #13 : Factorials

Example Question #14 : Factorials

Example Question #15 : Factorials

Example Question #16 : Factorials

Example Question #17 : Factorials

Example Question #18 : Factorials

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