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Algebra II : Factorials

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Example Question #91 : Factorials

Multiply: $[(2!)! \cdot (1-1)!]^2$

Possible Answers:

Correct answer:

Explanation:

Simplify all the terms in the parentheses first.

$(2!) = 2\times 1 =2$

(1-1)! = 0! = 1

This indicates that:

$(2!)! \cdot (1-1)! = 2\cdot 1 =2$

$[(2!)! \cdot (1-1)!]^2 = [2]^2 = 4$

The answer is:

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Example Question #91 : Factorials

Evaluate:

$\sum_{k=1}^{4} \frac{1}{k!}$

Possible Answers:

$1\frac{3}{4}$

$1\frac{17}{24}$

None of the other choices gives the correct response.

$1\frac{1}{33}$

$2 \frac{1}{12}$

Correct answer:

$1\frac{17}{24}$

Explanation:

$\sum_{k=1}^{4} \frac{1}{k!}$ is equal to the sum of the expressions formed by substituting 1, 2, 3 and 4, in turn, for in the expression $\frac{1}{k!}$ , as follows:

- or factorial - is defined to be the product of the integers from 1 to . Therefore, each term can be calculated by multiplying the integers from 1 to , then taking the reciprocal of the result.

k = 1 :

$\frac{1}{k!} = \frac{1}{1!} = \frac{1}{1} = 1$

k = 2 :

$\frac{1}{k!} = \frac{1}{2!} = \frac{1}{1 \cdot 2} = \frac{1}{2}$

k = 3 :

$\frac{1}{k!} = \frac{1}{3!} = \frac{1}{1 \cdot 2 \cdot 3} = \frac{1}{6}$

k = 4 :

$\frac{1}{k!} = \frac{1}{4!} = \frac{1}{1 \cdot 2 \cdot 3 \cdot 4} = \frac{1}{24}$

Add the terms:

$\sum_{k=1}^{4} \frac{1}{k!} = 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} = 1\frac{17}{24}$

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Example Question #91 : Factorials

Try without a calculator:

True or false: $\sqrt{9!}= 3!$

Possible Answers:

False

True

Correct answer:

False

Explanation:

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

$9! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 = 362,880$

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

Since $6^{2} = 36$ ,

$(3!)^{2} = 36 \ne 362,880 = 9!$

and

$\sqrt{9!}= 3!$ is a false statement.

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Example Question #94 : Factorials

Try without a calculator.

10,000 < N! < 100,000

Give the value of that makes this three-part inequality true.

Possible Answers:

The inequality has no solution.

N = 10

N = 7

N = 8

N = 9

Correct answer:

N = 8

Explanation:

Given a nonnegative integer , - or factorial - is defined to be the product of the integers from 1 to . After some exploration, multiplying 1 by 2, the result by 3, that result by 4, and so forth, we find that

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

Multiplying by 8, we find that

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

Multiplying by 9, we find that

$9! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9= 362,880$

Therefore,

7! < 10,000 < 8! < 100,000 < 9!

Since only integers may have factorial values, it follows that the only value of that makes the inequality true is N = 8 .

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Example Question #95 : Factorials

Try without a calculator:

is equal to which expression?

Possible Answers:

7! + 8

$7! \cdot 7$

None of these

7! + 7

$7! \cdot 8$

Correct answer:

$7! \cdot 8$

Explanation:

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

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

and

$8 ! = 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8$

$= (1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7) \cdot 8$

$= 7! \cdot 8$ ,

the correct response.

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Example Question #92 : Factorials

Try without a calculator:

Which is true about ?

Possible Answers:

0!=2

0! = 1

0! = 0

is an undefined quantity

Cannot be determined

Correct answer:

0! = 1

Explanation:

is considered to be equal to one by definition.

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Example Question #97 : Factorials

Try without a calculator:

Calculate:

(3!)!

Possible Answers:

Cannot be determined

362,880

120

720

Correct answer:

720

Explanation:

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

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

Therefore,

(3!)! = 6!

Since

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

it follows that 720 is the correct response.

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Algebra II : Factorials

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #91 : Factorials

Example Question #91 : Factorials

Example Question #91 : Factorials

Example Question #94 : Factorials

Example Question #95 : Factorials

Example Question #92 : Factorials

Example Question #97 : Factorials

All Algebra II Resources

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