Combinational Analysis

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GRE Quantitative Reasoning › Combinational Analysis

Questions 1 - 10

Mohammed is being treated to ice cream for his birthday, and he's allowed to build a three-scoop sundae from any of the thirty-one available flavors, with the only condition being that each of these flavors be unique. He's also allowed to pick different toppings of the available , although he's already decided well in advance that one of them is going to be peanut butter cup pieces.

Knowing these details, how many sundae combinations are available?

Explanation

Because order is not important in this problem (i.e. chocolate chip, pecan, butterscotch is no different than pecan, butterscotch, chocolate chip), it is dealing with combinations rather than permutations.

The formula for a combination is given as:

$C(n,k)=\frac{n!}{(n-k)!k!}$

where is the number of options and is the size of the combination.

For the ice cream choices, there are thirty-one options to build a three-scoop sundae. So, the number of ice cream combinations is given as:

$C(31,3)=\frac{31!}{(31-3)!3!}=4495$

Now, for the topping combinations, we are told there are ten options and that Mohammed is allowed to pick two items; however, we are also told that Mohammed has already chosen one, so this leaves nine options with one item being selected:

$C(9,1)=\frac{9!}{(9-1)!1!}=9$

So there are 9 "combinations" (using the word a bit loosely) available for the toppings. This is perhaps intuitive, but it's worth doing the math.

Now, to find the total sundae combinations—ice cream and toppings both—we multiply these two totals:

Daisy wants to arrange four vases in a row outside of her garden. She has eight vases to choose from. How many vase arrangements can she make?

Explanation

For this problem, since the order of the vases matters (red blue yellow is different than blue red yellow), we're dealing with permutations.

With selections made from potential options, the total number of possible permutations(order matters) is:

$P(n,k)=\frac{n!}{(n-k)!}$

$P(8,4)=\frac{8!}{(8-4)!}=1680$

Knowing these details, how many sundae combinations are available?

Explanation

The formula for a combination is given as:

$C(n,k)=\frac{n!}{(n-k)!k!}$

where is the number of options and is the size of the combination.

For the ice cream choices, there are thirty-one options to build a three-scoop sundae. So, the number of ice cream combinations is given as:

$C(31,3)=\frac{31!}{(31-3)!3!}=4495$

$C(9,1)=\frac{9!}{(9-1)!1!}=9$

So there are 9 "combinations" (using the word a bit loosely) available for the toppings. This is perhaps intuitive, but it's worth doing the math.

Now, to find the total sundae combinations—ice cream and toppings both—we multiply these two totals:

Evaluate:

Explanation

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

There are 12 boys in a football competition, the top 3 competitors are awarded with an trophy. How may possible groups of 3 are there for this competition?

Explanation

This is a permutation. A permutation is an arrangement of objects in a specific order.

The formula for permutations is:

$_{n}P_{k} = \frac{n!}{(n-k)!} - n(n-1) (n-2)...(n-k+1)$

This is written as $_{n}P_{k}$

$_{12}P_{3} = 12\times11\times10 = 1,320$

There are possible groups of 3.

Evaluate:

Explanation

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

Daisy wants to arrange four vases in a row outside of her garden. She has eight vases to choose from. How many vase arrangements can she make?

Explanation

For this problem, since the order of the vases matters (red blue yellow is different than blue red yellow), we're dealing with permutations.

With selections made from potential options, the total number of possible permutations(order matters) is:

$P(n,k)=\frac{n!}{(n-k)!}$

$P(8,4)=\frac{8!}{(8-4)!}=1680$

There are 12 boys in a football competition, the top 3 competitors are awarded with an trophy. How may possible groups of 3 are there for this competition?

Explanation

This is a permutation. A permutation is an arrangement of objects in a specific order.

The formula for permutations is:

$_{n}P_{k} = \frac{n!}{(n-k)!} - n(n-1) (n-2)...(n-k+1)$

This is written as $_{n}P_{k}$

$_{12}P_{3} = 12\times11\times10 = 1,320$

There are possible groups of 3.

An ice cream shop has 23 flavors. Melissa wants to buy a 3-scoop cone with 3 different flavors, How many cones can she buy if order is important?

Explanation

This is a permutation. A permutation is an arrangement of objects in a specific order.

The formula for permutations is:

$_{n}P_{k} = \frac{n!}{(n-k)!} - n(n-1) (n-2)...(n-k+1)$

This is written as $_{n}P_{k}$

$_{23}P_3=P(23,3)$ represents the number of permutations of 23 things taken 3 at a time.

$P(23,3) = 23\times22\times21 = 10,626$

Simplify the following expression:

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

Explanation

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:

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