Factorials - GRE Quantitative Reasoning

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Question

Simplify the following expression:

$\frac{12!}{10!-4!}$

Answer

Recall that ! means facotrial in math. This means we multiply the number by all positive integers less than itself. In other words, this...

$\frac{12!}{10!-4!}$

Becomes

$\frac{1\cdot 2\cdot3\cdot 4\cdot5\cdot6\cdot7\cdot8\cdot9\cdot10\cdot11\cdot12}{(1\cdot2\cdot3\cdot4\cdot5\cdot6\cdot7\cdot8\cdot9\cdot10)-(1\cdot2\cdot3\cdot 4)}$

This is a great job for a calculator, which yields:

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Question

Evaluate:

Answer

Step 1: We need to define factorial (!). The operation (!) means that you multiply the number next to the ! by the next consecutive number down to 1.

Step 2: Expand the equation:

$5!=5\cdot4\cdot3\cdot2\cdot1=120$
$3(4\cdot 3\cdot 2\cdot 1)=3(24)=72$

Step 3: Evaluate the expression:

The answer to the expression is .

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Question

$\frac{5!}{3!-2!}$

Answer

is the product of consecutive numbers 1 through

$\frac{5\times 4\times 3\times 2\times 1}{(3\times 2\times 1)-(2\times 1)} =$

$\frac{120}{6-2} = \frac{120}{4} = 30$

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Question

Answer

is the product of consecutive numbers 1 through

$(6\times 5\times 4\times 3\times 2\times 1) - 2(3\times 2\times 1) =$

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Question

What is $_{8}P_{4}$ equal to?

Answer

depicts the permutation of n total objects with k objects being arranged.

Thus,

$_{8}P_{4} = \frac{n!}{k!}=\frac{8!}{4!}$

The upper factorial is the upper index, and the lower factorial is the difference of the indices.

$\frac{8!}{4!} = \frac{8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{4\times 3\times 2\times 1}$

The or $4\times3\times2\times1$ will cancel out.

$8\times7\times6\times5 = 56\times30 = 1,680$

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Question

Twelve horses run a race. In how many ways can 3 horses finish in 1st, 2nd, and 3rd place, in any order?

Answer

This particular question is asking one to find the combination. This is because the order of the 1st, 2nd, and 3rd place finishers is not important.

Thus evaluate $_{12}C_{3}$ .

$_{12}C_{3} = \frac{12!}{(12-3)! 3!}$

Because and both consist of , the will cancel out leaving:

$\frac{12\times 11\times 10}{3\times 2\times 1}$

$\frac{12\times 11\times 10}{3\times 2\times 1} = \frac{1320}{6} = 220$

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Question

Evaluate: $\frac {13!}{10!}$

Answer

Step 1: Rewrite the equation by using rule of factorials...

Rule of factorials: Multiply the number in front of factorial and each number one lower than the rest until I hit .

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

Step 2: Reduce any common numbers (colored in blue):

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

We get $13\cdot12\cdot11$ !

Step 3: Evaluate what's left

$13\cdot12\cdot11=1716$

The value of the fraction is .

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Question

Evaluate:

Answer

Step 1: Evaluate

$10!=10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1=3628800$

Step 2: Evaluate

$4(9!)=4(9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1)=1451520$

Step 3: Evaluate

$8!=8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1=40320$

Step 4: Evaluate

$3(7!)=3(7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1)=15120$

Step 5: Add/Subtract values:

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Question

Evaluate: $3\cdot \bigg[\frac {9!}{7!}\bigg]$

Answer

Step 1: Do the division of the factorials

$\bigg[\frac {9!}{7!}\bigg]=\frac {9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}$

Step 2: Simplify by cancelling out any terms on both top and bottom:

$\frac {9\cdot 8\cdot {\color{Blue} 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}}{{\color{Blue} 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}}=9\cdot8=72$

Step 3: Multiply the result of the division by 3..

$72 \cdot 3=216$

The value of that expression is .

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Question

Evaluate:

Answer

Step 1: Find

$7!=7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1=5040$

Step 2: Find

$6(6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1)=6(720)=4320$

Step 3: Subtract the values from step 1 and step 2:

The answer is

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Question

Evaluate:

Answer

Step 1: Write in multiplication form:

$14!=14\cdot13\cdot12\cdot11\cdot10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1$

Step 2: Multiply Out

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Question

Evaluate: $\frac {14!}{13!}$

Answer

Step 1: Expand the numerator and denominator:

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

Step 2: Cross all numbers out that are common to both the numerator and denominator:

We are left with .

The value of $\frac {14!}{13!}=14$

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Question

What is ?

Answer

Step 1: Recall definition of factorial...

Factorial is a mathematical expression where you multiply by every number below it until I hit . To find each number, subtract from the previous number

Step 2: Write out

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Question

Evaluate:

Answer

Step 1: Evaluate
Factorial-Multiply the number in front of the exclamation point by every number below until I hit

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Question

Evaluate:

Answer

Step 1: Evaluate each factorial expression separately...

Step 2: Take the evaluation of each expression and apply the operations in the problem:

So, we get

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Question

Evaluate: $3!^{2^{3!}}$

Answer

Step 1: Apply the power rule (stacked) of exponents..

We get $3!^{23!}$

Step 2: Evaluate the exponent...

We get $3!^{26}=3!^{12}$

Step 3: Re-write the base:

Final answer: $6^{12}$

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Question

Evaluate: $\large 4(3!)$

Answer

Step 1: Evaluate the parentheses first

$\large (3!)=(321)=6$

Step 2: Multiply the outside by the final value in Step 1..

$\large 6(4)=24$

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Question

Evaluate: $\large 5!$

Answer

Step 1: Define Factorial

Factorial is a mathematical expression which you multiply the number in front of the exclamation point by every number below it until you hit 1..

Step 2: Evaluate $\large 5!$

$\large 5!=5(5-1)(5-2)(5-3)(5-4)$
$\large 5!=54321=120$

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Question

Evaluate:

Answer

Evaluate $\large 7!$ :

$\large 7!=7\cdot 6\cdot5\cdot4\cdot3\cdot2\cdot1=5040$

Evaluate $\large 6(6!)$ :

$\large \large 6(6!)=6(6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1)=6(720)=4320$

Evaluate $\large 7!-6(6!)$ :

$\large 5040-4320=720$

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Question

Simplify the following expression:

$\frac{12!}{10!-4!}$

Answer

Recall that ! means factorial in math. This means we multiply the number by all positive integers less than itself. In other words, this...

$\frac{12!}{10!-4!}$

Becomes

$\frac{1\cdot 2\cdot3\cdot 4\cdot5\cdot6\cdot7\cdot8\cdot9\cdot10\cdot11\cdot12}{(1\cdot2\cdot3\cdot4\cdot5\cdot6\cdot7\cdot8\cdot9\cdot10)-(1\cdot2\cdot3\cdot 4)}$

This is a great job for a calculator, which yields:

Compare your answer with the correct one above

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