Solve ((17 - n)!)/((14 - n)!) = 5!

Question

Solve the equation

n = 11

Evaluate

\frac{( 17 - n ) !}{( 14 - n ) !} = 5!

Find the domain

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Evaluate

⎩ ⎨ ⎧ (14 - n)! \neq = 0 17 - n \in N 14 - n \in N

Calculate

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Evaluate

(14 - n)! \neq = 0

Rewrite the expression

1 \neq = 0

The statement is true for any value of n

n \in R

⎩ ⎨ ⎧ n \in R 17 - n \in N 14 - n \in N

Find the intersection

⎩ ⎨ ⎧ 14 - n \in N 17 - n \in N n \in R

Calculate

14 - n \in N, 17 - n \in N, n \in R

\frac{( 17 - n ) !}{( 14 - n ) !} = 5!, 14 - n \in N, 17 - n \in N, n \in R

Divide the terms

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Evaluate

\frac{( 17 - n ) !}{( 14 - n ) !}

Factor the expression

\frac{( 14 - n ) ! \times ( 17 - n ) ( 16 - n ) ( 15 - n )}{( 14 - n ) !}

Reduce the fraction

(17 - n) (16 - n) (15 - n)

(17 - n) (16 - n) (15 - n) = 5!

Evaluate

(17 - n) (16 - n) (15 - n) = 120

Expand the expression

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Evaluate

(17 - n) (16 - n) (15 - n)

Multiply the terms

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Evaluate

(17 - n) (16 - n)

Apply the distributive property

17 \times 16 - 17 n - n \times 16 - (- n \times n)

Multiply the numbers

272 - 17 n - n \times 16 - (- n \times n)

Use the commutative property to reorder the terms

272 - 17 n - 16 n - (- n \times n)

Multiply the terms

272 - 17 n - 16 n - (- n^{2})

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

272 - 17 n - 16 n + n^{2}

Subtract the terms

272 - 33 n + n^{2}

(272 - 33 n + n^{2}) (15 - n)

Apply the distributive property

272 \times 15 - 272 n - 33 n \times 15 - (- 33 n \times n) + n^{2} \times 15 - n^{2} \times n

Multiply the numbers

4080 - 272 n - 33 n \times 15 - (- 33 n \times n) + n^{2} \times 15 - n^{2} \times n

Multiply the numbers

4080 - 272 n - 495 n - (- 33 n \times n) + n^{2} \times 15 - n^{2} \times n

Multiply the terms

4080 - 272 n - 495 n - (- 33 n^{2}) + n^{2} \times 15 - n^{2} \times n

Use the commutative property to reorder the terms

4080 - 272 n - 495 n - (- 33 n^{2}) + 15 n^{2} - n^{2} \times n

Multiply the terms

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Evaluate

n^{2} \times n

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

n^{2 + 1}

Add the numbers

n^{3}

4080 - 272 n - 495 n - (- 33 n^{2}) + 15 n^{2} - n^{3}

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

4080 - 272 n - 495 n + 33 n^{2} + 15 n^{2} - n^{3}

Subtract the terms

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Evaluate

- 272 n - 495 n

Collect like terms by calculating the sum or difference of their coefficients

(- 272 - 495) n

Subtract the numbers

- 767 n

4080 - 767 n + 33 n^{2} + 15 n^{2} - n^{3}

Add the terms

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Evaluate

33 n^{2} + 15 n^{2}

Collect like terms by calculating the sum or difference of their coefficients

(33 + 15) n^{2}

Add the numbers

48 n^{2}

4080 - 767 n + 48 n^{2} - n^{3}

4080 - 767 n + 48 n^{2} - n^{3} = 120

Move the expression to the left side

4080 - 767 n + 48 n^{2} - n^{3} - 120 = 0

Subtract the numbers

3960 - 767 n + 48 n^{2} - n^{3} = 0

Factor the expression

(11 - n) (360 - 37 n + n^{2}) = 0

Separate the equation into 2 possible cases

11 - n = 0 360 - 37 n + n^{2} = 0

Solve the equation

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Evaluate

11 - n = 0

Move the constant to the right-hand side and change its sign

- n = 0 - 11

Removing 0 doesn't change the value,so remove it from the expression

- n = - 11

Change the signs on both sides of the equation

n = 11

n = 11 360 - 37 n + n^{2} = 0

Solve the equation

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Evaluate

360 - 37 n + n^{2} = 0

Rewrite in standard form

n^{2} - 37 n + 360 = 0

Substitute a= 1,b= - 37 and c= 360 into the quadratic formula n = \frac{- b \pm b ^{2} - 4 a c}{2 a}

n = \frac{37 \pm ( - 37 ) ^{2} - 4 \times 360}{2}

Simplify the expression

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Evaluate

(- 37)^{2} - 4 \times 360

Multiply the numbers

(- 37)^{2} - 1440

Rewrite the expression

3 7^{2} - 1440

Evaluate the power

1369 - 1440

Subtract the numbers

- 71

n = \frac{37 \pm - 71}{2}

The expression is undefined in the set of real numbers

n \in / R

n = 11 n \in / R

Find the union

n = 11

Check if the solution is in the defined range

n = 11, 14 - n \in N, 17 - n \in N, n \in R

Solution

n = 11

Show Solution