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Algebra II : Factorials

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Example Question #8 : Other Factorials

What is ?

Possible Answers:

Correct answer:

Explanation:

A factorial means you are multiplying the number, with one less than previous number and you keep multiplying until you reach .

So, is $3\cdot 2\cdot 1$ .

Multiplying that out, we get .

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Example Question #9 : Other Factorials

What is ?

Possible Answers:

120

100

Correct answer:

120

Explanation:

A factorial means you are multiplying the number, with one less than previous number and you keep multiplying until you reach .

So, is $5\cdot 4\cdot 3\cdot 2\cdot 1$ .

Multiplying that out, we get 120 .

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Example Question #10 : Other Factorials

What is $3! \cdot 4!$ ?

Possible Answers:

5040

144

256

Correct answer:

144

Explanation:

A factorial means you are multiplying the number, with one less than previous number and you keep multiplying until you reach .

So, is $3\cdot 2\cdot 1$ . Multiplying that out, we get .

We also have . That is $4\cdot 3\cdot 2\cdot 1$ or .

Since we are multiplying the factorial, we multiply and to get 144 .

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Example Question #81 : Factorials

What is $\frac{6!}{4!}$ ?

Possible Answers:

$\frac{6}{5}$

Correct answer:

Explanation:

A factorial means you are multiplying the number, with one less than previous number and you keep multiplying until you reach .

So, is $6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1$ . We also have . That is $4\cdot 3\cdot 2\cdot1$ .

Since we are dividing the factorial, we can cancel out some terms.

Both the numerator and denominator have $4\cdot 3\cdot 2\cdot 1$ , and we can cancel those out. We are left with $6\cdot 5$ or 30.

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Example Question #11 : Other Factorials

What is $\frac{7!}{30}$ ?

Possible Answers:

$\frac{1}{168}$

$\frac{1}{7}$

168

Correct answer:

168

Explanation:

A factorial means you are multiplying the number, with one less than previous number and you keep multiplying until you reach .

So, is $7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1$ . We are dividing by 30. With careful inspection, can be broken down to $6\cdot5$ .

If we cancel that out with the ones in the numerator, we have $7\cdot 4\cdot 3\cdot 2\cdot 1$ or 168 .

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Example Question #83 : Factorials

What is $\frac{x!}{(x+1)!}$ ?

Possible Answers:

x+1

$\frac{1}{x+1}$

x^2+x

Correct answer:

$\frac{1}{x+1}$

Explanation:

A factorial means you are multiplying the number, with one less than previous number and you keep multiplying until you reach . Since we are dealing with variables, let's analyze each the numerator and the denominator. x+1 is definitely greater than in this situation because factorials are always positive numbers. If we took the difference between x+1 and we would get . This means that if we were to expand both the numerator and denominator, we cancel everything out except the extra term in the denominator which is x+1 .

$\frac{x!}{(x+1)!)}=\frac{x\cdot 1}{(x+1)(x)(1)}=\frac{1}{x+1}$

So final answer is $\frac{1}{x+1}$ .

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Example Question #84 : Factorials

Simplify.

$\frac{(x+2)!}{(x-2)!}$

Possible Answers:

x(x-1)

(x+1)x(x-1)

(x+2)(x+1)x

(x+2)(x+1)

(x+2)(x+1)x(x-1)

Correct answer:

(x+2)(x+1)x(x-1)

Explanation:

A factorial means you are multiplying the number, with one less than previous number and you keep multiplying until you reach . Since we are dealing with variables, let's analyze each the numerator and the denominator. x+2 is definitely greater than x-2 in this situation because factorials are always positive numbers. If we took the difference between x+2 and x-2 we would get . This means that if we were to expand both the numerator and denominator, we cancel everything out except the extra terms in the numerator which are x+2, x+1, x, x-1 . So final answer is (x+2)(x+1)x(x-1) .

If you need convincing, let x=6 . So we have $\frac{(6+2)!}{(6-2)!}=\frac{8!}{4!}=\frac{8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{4\cdot 3\cdot 2\cdot 1}$ .

You can cancel out the $4\cdot 3\cdot 2\cdot1$ from top and bottom to get $8\cdot 7\cdot 6\cdot 5$ . We need to express them into expressions.

Since was , to get , you need to add to or x+2.

To get , you add to or x+6.

x=6 To get you subtract from or x-1 .

You still get the same answer of (x+2)(x+1)x(x-1) .

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Example Question #85 : Factorials

Simplify: $\frac{2! + 0!}{3!-0!}$

Possible Answers:

$\frac{1}{5}$

$\frac{3}{4}$

$\frac{2}{3}$

$\frac{1}{2}$

$\frac{3}{5}$

Correct answer:

$\frac{3}{5}$

Explanation:

Write out the terms of each factorial in expanded form. The only exception is the zero factorial, which is equal to one.

$\frac{2! + 0!}{3!-0!} =\frac{2\times 1+1}{3\times 2\times 1-1}$

Simplify the terms. The values in the numerator and denominator cannot cancel out!

$\frac{2\times 1+1}{3\times 2\times 1-1} = \frac{2+1}{6-1} = \frac{3}{5}$

The correct answer is: $\frac{3}{5}$

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Example Question #86 : Factorials

What is the value of ?

Possible Answers:

40320

5040

3268800

69120

Correct answer:

40320

Explanation:

Step 1: We must define factorial. Factorial is defined as , where n!=n(n-1)(n-2)(n-3)...1

Step 2: Evaluate

8!=8(8-1)(8-2)(8-3)(8-4)(8-5)(8-6)(8-7)
$8!=8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1=40320$

$40320\equiv 8!$

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Example Question #87 : Factorials

Compute: $\frac{(3!)!+3(3!)}{3!}$

Possible Answers:

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

345

123

246

456

Correct answer:

123

Explanation:

Simplify the factorials in the numerator and denominator.

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

Simplify the terms on the top and bottom.

$\frac{(3 \times 2 \times 1)!+3(3 \times 2 \times 1)}{3 \times 2 \times 1} =\frac{(6)!+3(6)}{6}$

$\frac{(6)!+3(6)}{6} = \frac{(6\times 5\times4\times3\times2\times1)+3(6)}{6}$

$\frac{720+18}{6} = \frac{738}{6} = 123$

The answer is: 123

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Algebra II : Factorials

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #8 : Other Factorials

Example Question #9 : Other Factorials

Example Question #10 : Other Factorials

Example Question #81 : Factorials

Example Question #11 : Other Factorials

Example Question #83 : Factorials

Example Question #84 : Factorials

Example Question #85 : Factorials

Example Question #86 : Factorials

Example Question #87 : Factorials

All Algebra II Resources

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