Counting Ii Statistics Worksheet printable pdf download

Counting Ii Statistics Worksheet

Counting II

Sometimes we will want to choose k objects from a set of n objects, and

we won’t be interested in ordering them. For example, if you are leaving for

vacation and you want to pack your suitcase with three of the seven pairs

of shorts that you own, then it doesn’t matter in which order you pack the

shorts. All that matters is which three pairs you pack.

n choose k

The number of di↵erent ways that k objects can be chosen from a set of

n objects (when order doesn’t matter) is called n choose k. It is written in

symbol form as

Examples.

• There are four di↵erent ways that one letter can be chosen from the

set of four letters {e, f , g, a}. One way is to choose the letter e. Alternatively,

you could also choose the letter f , or the letter g, or the letter a.

Since there are 4 options for choosing one object from a set of 4 objects,

we have

= 4.

• Below is a list of all the possible ways that 2 numbers can be chosen

from the set of four numbers {3, 7, 2, 9}. There are six di↵erent ways. Thus,

= 6.

3, 7

3, 2

3, 9

7, 2

7, 9

2, 9

General formula

To say that we are choosing and ordering k objects from a set of n objects

is to say that we are performing 2 separate tasks. First is the task of choosing

k objects from the set of n objects, and the number of ways to perform that

task is

. Second is the task of ordering the k objects after we’ve chosen

them. There are k! ways to order k objects.

Let’s repeat that. To choose and order k objects: First, choose the k

objects, then order the k objects you chose. Options multiply, so the total

number of ways that we can choose and order k objects from a set of n objects

k!.

Counting Ii Statistics Worksheet