Permutations

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GRE Quantitative Reasoning › Permutations

Questions 1 - 10

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.

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

people are at a farewell party. At the end of the night, each person shakes hands. How many handshakes are made?

NOTE: No two people can shake hands more than once.

Explanation

Step 1: Determine how many people are there..
There are people.
Step 2: Determine how many people shake hands in a handshake...
people make one handshake.
Step 3: Determine how many handshakes can be made...
We have a restriction here, so we need to use permutation..

So, there will be handshakes.

$11P2=\frac {11*10}{2*1}=11*5=55$ handshakes

Evaluate .

Explanation

is asking to find the permutation of four items when you want to choose all four. When dealing with permutations, order matters.

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

The formula for permutations in this case will be,

or factorial.

$P(4,4) = 4\times3\times2\times1 = 24$

There are people at a family dinner. After the dinner is over, people shake hands with each other. How many handshakes were there between these people. Note: Once two people shake hands, they cannot shake hands again..

Explanation

Step 1: A handshake MUST ALWAYS be between TWO people.

Step 2: Break down each person and who they can shake hands with:

Person can shake hands with: .
Person can shake hands with:
Person can shake hands with:
Person can shake hands with:
Person can shake hands with:
Person can shake hands with:
Person can shake hands with:
Person can shake hands with:
Person can shake hands with:
Person can shake hands with:
Person already shook everybody's hand..

Step 3: Count how many handshakes each person can make:

Person shakes hands times.
Person shakes hands times.
Person shakes hands times.
Person shakes hands times.
Person shakes hands times.
Person shakes hands times.
Person shakes hands times.
Person shakes hands times.
Person shakes hands times.
Person shakes hands time.
Person already shook everybody's hand.

Step 4: Add up the number of times each person shook hands:

There were handshakes made between these people.

Quantity A: The number of possible permutations when seven choices are made from ten options.

Quantity B: The number of possible permutations when five choices are made from eleven options.

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined.

Explanation

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

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

Quantity A:

$P(10,7)=\frac{10!}{(10-7)!}=604800$

Quantity B:

$P(11,5)=\frac{11!}{(11-5)!}=55440$

Quantity A is greater.

Lisa is dressing warm for the winter. She'll be layering three shirts over each other, and two pairs of socks. If she has fifteen shirts to choose from, along with ten different kinds of socks, how many ways can she layer up?