Algebra II - Multiplying and Dividing Factorials

Divide: $\frac{3!}{5!}\div 3!$

$\frac{1}{120}$

$\frac{3}{10}$

$\frac{1}{40}$

$\frac{3}{40}$

$\frac{1}{60}$

Explanation

Evaluate by expanding the terms of the factorials.

$\frac{3!}{5!}\div 3! = \frac{3\times2\times1 }{5\times4\times3\times2\times1 }\div (3\times2\times1 )$

Cancel out the common terms in the first fraction.

$\frac{1 }{5\times4 }\div (3\times2\times1 )$

Change the division sign to a multiplication and take the reciprocal of the second quantity.

$\frac{1 }{5\times4 }\times \frac{1}{(3\times2\times1 )} =\frac{1}{120}$

The answer is: $\frac{1}{120}$

Stewie has $\frac{5!}{3!}$ marbles in a bag. How many marbles does Stewie have?

Explanation

Simplifying this equation we notice that the 3's, 2's, and 1's cancel so

Alternative Solution

Find the value of:

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

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:

$7\times6\times5\times4 = 840$

$\frac{15!}{13!}$

Explanation

When you have factorials in the numerator and denominator of a fraction, you can cancel out the common factors between them. It can help to write the problems in expanded form.

$\frac{15!}{13!}$

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

The entire denominator 13 will be cancelled out leaving only

$15\cdot 14$

Multiply: $2! \cdot 4!$

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:

$\frac{7!}{4!}\cdot \frac{5!}{2!}$

Explanation

Similar factors that are in both the numerator and denominator of a fraction cancel out. It helps to write it in expanded form.

$\frac{7!}{4!}\cdot \frac{5!}{2!}$

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

$7\cdot 6\cdot 5 \cdot 5\cdot 4\cdot 3$

Simplify the factorials: $(7-4)!\cdot(7!-5!)$

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

Explanation

Evaluate the first term. Write out the terms of the factorial.

$(7-4)! = 3! = 3\times 2 \times 1 = 6$

Evaluate the second term.

$7! = 7\times6\times5\times4\times3\times2\times1 = 5040$

$5!=5\times4\times3\times2\times1 = 120$

Combine the simplified terms.

$(7-4)!\cdot(7!-5!) = 6\cdot 4920$

The answer is:

Simplify: $\frac{2(3!+4)}{(2+2)!}$

$\frac{5}{6}$

$\frac{181}{6}$

$\frac{10}{3}$

$\frac{91}{3}$

Explanation

Simplify the denominator.

$\frac{2(3!+4)}{(2+2)!}=\frac{2(3!+4)}{4!}$

Rewrite each factorial in the given fraction. Do not distribute the two through the quantity or this will change the value of the factorial.

$\frac{2(3!+4)}{4!} =\frac{2([3\times 2\times 1]+4)}{[4\times3\times 2\times 1] } = \frac{2(6+4)}{24}=\frac{2(10)}{24}=\frac{20}{24}$

Reduce this fraction.

The answer is: $\frac{5}{6}$

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

$\textup{Undefined}$

Explanation

Evaluate the factorials first.

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

Substitute the values back into the expression.

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

Simplify the numerator.

Simplify the following factorial:

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

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...

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:

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

Multiplying and Dividing Factorials

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