Factorials - Algebra II

Card 0 of 388

Question

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

Answer

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

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

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

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

Compare your answer with the correct one above

Question

$\frac{6!\cdot 4!\cdot 2!}{10!}$

Answer

Common factors on the numerator and denominator of a fraction will cancel. It helps to write problems in expanded form.

$\frac{6!\cdot 4!\cdot 2!}{10!}$

$\frac{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1\cdot 4\cdot 3\cdot 2\cdot 1\cdot 2\cdot 1}{10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}$

$\frac{4\cdot 3\cdot 2\cdot 1\cdot 2\cdot 1}{10\cdot 9\cdot 8\cdot 7}$

$\frac{48}{5040}$

Then, reduce the fraction to lowest terms. Both numerator and denominator divide by 48.

$\frac{1}{105}$

Compare your answer with the correct one above

Question

$\frac{4!}{3!\cdot 2!}$

Answer

Common factors in the numerator and denominator will cancel. It helps to write factorials in expanded form.

$\frac{4!}{3!\cdot 2!}$

$\frac{4\cdot 3\cdot 2\cdot 1}{3\cdot 2\cdot 1\cdot 2\cdot 1}$

$\frac{4}{2\cdot 1}$

$\frac{4}{2}$

Compare your answer with the correct one above

Question

$5!\cdot \frac{4!}{2!}\cdot \frac{2!}{6!}$

Answer

First, simplify the factorials in each fraction by canceling common factors in the numerator and denominator. It can help to write it in expanded form.

$5!\cdot \frac{4!}{2!}\cdot \frac{2!}{6!}$

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

It is easiest to combine these numbers into one fraction. The 5! will go into the numerator

$\frac{5\cdot 4\cdot 3\cdot 2\cdot 1\cdot 4\cdot 3\cdot 2\cdot 1\cdot 2\cdot 1}{2\cdot 1\cdot 6\cdot5\cdot 4\cdot 3\cdot 2\cdot 1 }$

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

$\frac{24}{6}$

Compare your answer with the correct one above

Question

$\frac{3!}{5!}\cdot \frac{6!}{7!}\div \frac{8!}{6!}$

Answer

Simplify the factorials in each fraction by canceling common factors in the numerator and denominator. It can help to write it in expanded form.

$\frac{3!}{5!}\cdot \frac{6!}{7!}\div \frac{8!}{6!}$

$\frac{3\cdot 2\cdot 1}{5\cdot 4\cdot 3\cdot 2\cdot 1}\cdot \frac{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}\div \frac{8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}$

$\frac{1}{5\cdot 4}\cdot \frac{1}{7}\div \frac{8\cdot 7}{1}$

$\frac{1}{20}\cdot \frac{1}{7}\div \frac{56}{1}$

$\frac{1}{140}\div \frac{56}{1}$

$\frac{1}{140}\cdot \frac{1}{56}$

$\frac{1}{7840}$

Compare your answer with the correct one above

Question

$\frac{20!\cdot 10!}{25!\cdot 5!}$

Answer

Simplify the factorials the fraction by canceling common factors in the numerator and denominator. Normally, it would help to write out the factorials in expanded form, but since these are larger factorials that is not feasible. The best approach to cancelling these is thinking that 20! will be completely gone by taking it out of 25! and 5! will be completely gone by taking it out of 10!

$\frac{20!\cdot 10!}{25!\cdot 5!}$

$\frac{10\cdot 9\cdot 8\cdot 7\cdot 6}{25\cdot 24\cdot 23\cdot 22\cdot 21}$

$\frac{30240}{6375600}$

Then reduce this to lowest terms.

$\frac{3024}{637560}=\frac{1512}{318780}=\frac{756}{159390}=\frac{378}{79695}=\frac{54}{11385}=\frac{6}{1265}$

Compare your answer with the correct one above

Question

$\frac{30!\cdot 2!}{30!}$

Answer

Simplify the factorials in each fraction by canceling common factors in the numerator and denominator. It would be helpful to write this in expanded form but the factorials are too large for this to be feasible.

$\frac{30!\cdot 2!}{30!}$

$\frac{2\cdot 1}{1}$

$\frac{2}{1}$

Compare your answer with the correct one above

Question

What is the value of ?

Answer

A factorial represents the product of all natural numbers less than a given number. Thus, which gives us .

Compare your answer with the correct one above

Question

$\frac{2!}{13!}\div \frac{3!}{15!}$

Answer

Simplify the factorials in each fraction by canceling common factors in the numerator and denominator. It can help to write it in expanded form.

$\frac{2!}{13!}\div \frac{3!}{15!}$

$\frac{2\cdot 1}{13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}\div \frac{3\cdot 2\cdot 1}{15\cdot 14\cdot13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1 }$

$\frac{1}{13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3}\div \frac{1}{15\cdot 14\cdot13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4}$

Since it is division of fractions change the division sign to multiplication and flip the second fraction

$\frac{1}{13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3}\cdot \frac{15\cdot 14\cdot13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4}{1}$

$\frac{15\cdot 14\cdot13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4}{13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3}$

$\frac{15\cdot 14}{3}$

$\frac{210}{3}$

Compare your answer with the correct one above

Question

$\frac{8!\cdot 7!\cdot 3!}{9!\cdot 7!}$

Answer

Simplify the factorials in the fraction by canceling common factors in the numerator and denominator. It can help to write it in expanded form, but I would suggest cancelling out the 7! in the numerator and denominator first.

$\frac{8!\cdot 7!\cdot 3!}{9!\cdot 7!}$

$\frac{8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1\cdot 3\cdot 2\cdot 1}{9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}$

$\frac{3\cdot 2\cdot 1}{9}$

$\frac{6}{9}$

Then reduce the fraction to lowest terms

$\frac{2}{3}$

Compare your answer with the correct one above

Question

$\frac{4!}{6!}\cdot \frac{7!}{2!}\cdot 2!$

Answer

Simplify the factorials in the fraction by canceling common factors in the numerator and denominator. It can help to write it in expanded form.

$\frac{4!}{6!}\cdot \frac{7!}{2!}\cdot 2!$

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

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

Combine each of the fractions into one fraction.

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

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

$\frac{168}{1}$

Compare your answer with the correct one above

Question

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

Answer

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

Alternative Solution

Compare your answer with the correct one above

Question

Which of the following is equivalent to $6!\times4!$ ?

Answer

This is a factorial question. The formula for factorials is $n! = n\times(n-1)\times(n-2)\times...\times1$ .

$6!=6\times5\times4! = 30\times 4!$

Compare your answer with the correct one above

Question

Which of the following is NOT the same as $\frac{6!\times5!}{4!}$ ?

Answer

The cancels out all of except for the parts higher than 4, this leaves a 6 and a 5 left to multilpy

Compare your answer with the correct one above

Question

Find the value of:

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

Answer

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:

$7\times6\times5\times4 = 840$

Compare your answer with the correct one above

Question

Simplify the following expression:

$\frac{7!(n+3)!}{6!(n+2)!}$

Answer

Recall that .

Likewise, $6! = 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1$ .

Thus, the expression $\frac{7!(n+3)!}{6!(n+2)!}$ can be simplified in two parts:

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

and

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

The product of these two expressions is the final answer:

Compare your answer with the correct one above

Question

$\frac{6!}{2!4!} = ?$

Answer

To simplify this, just write out each factorial:

$\frac{6!}{2!4!} = \frac{654321}{(21)(4321)} = \frac{65}{2*1} = 15$

Compare your answer with the correct one above

Question

$\frac{\left ( n+2\right )!}{n!}$

If is a postive integer, which of the following answer choices is a possible value for the expression.

Answer

This expression of factorials reduces to (n+1)(n+2). Therefore, the solution must be a number that multiplies to 2 consecutive integers. Only 30 is a product of 2 consecutive integers. $5\ast6=30$

So n would have to be 4 in this problem.

Compare your answer with the correct one above

Question

Divide by

Answer

A factorial is a number which is the product of itself and all integers before it. For example $5!=5\cdot 4 \cdot 3 \cdot 2 \cdot 1$

In our case we are asked to divide by . To do this we will set up the following:

$\frac{2015!}{2014!}$

We know that can be rewritten as the product of itself and all integers before it or:

$2015!=2015 \cdot 2014!$

Substituting this equivalency in and simplifying the term, we get:

$\frac{2015 \cdot 2014!}{2014!}=2015$

Compare your answer with the correct one above

Question

Simplify:

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

Answer

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:

$\frac{20!}{17!\times 2!}=\frac{20\cdot19\cdot18\cdot17\cdot16\cdot15\cdot14\cdot13\cdot12\cdot11\cdot10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}{17\cdot16\cdot15\cdot14\cdot13\cdot12\cdot11\cdot10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1\cdot 2}$

$=\frac{20\cdot19\cdot18}{2}=10\cdot19\cdot18=3420$

Compare your answer with the correct one above

Tap the card to reveal the answer