Counting Ii Statistics Worksheet (Page 5 of 9) in pdf

a set of n objects is the same as leaving n

k. Therefore,

This formula tells us that the rows in Pascal’s triangle will read the same

left-to-right as they will right-to-left. You can see that in the triangle on the

next page. What we saw earlier in the formula

was the special

n 1

case of the formula

when k = n

n k

Add the two numbers above to get the number below

Suppose that you have a set of n di↵erent rocks: 1 big red brick, and n

di↵erent little blue marbles. How many di↵erent ways are there to choose

k + 1 rocks from the set of n rocks?

Any collection of k + 1 rocks either includes the big red brick, or it doesn’t.

Let’s ﬁrst look at those collections of k + 1 objects that do contain a big

red brick. One of the k + 1 objects we will choose is a big red brick. That’s

a given. That means that all we have to do is decide which k of the little

blue marbles we want to choose along with the big red brick to make up our

n 1

collection of k + 1 objects. There are

di↵erent ways we could choose k

marbles from the total number of n

1 little blue marbles. Thus, there are

n 1

di↵erent ways we could choose a set of k + 1 objects from our set of n

rocks if we know that one of the objects we will choose is a big red brick.

Now let’s look at those collections of k +1 objects that don’t contain the big

red brick. Then all k + 1 objects that we will choose are little blue marbles.

There are n 1 little blue marbles, and the number of di↵erent ways we could

n 1

choose k + 1 of the n

1 little blue marbles is

k+1

Any collection of k + 1 rocks either includes the big red brick, or it doesn’t.

So to ﬁnd the number of ways that we could choose k + 1 objects, we just

have to add the number of possibilities that contain a big red brick, to the

number of possibilities that don’t contain a big red brick. That formula is

k + 1

If n = 6 and k = 2, then the above formula says that

Looking at Pascal’s triangle, you’ll see that

and

are the two numbers

that are just above the number

Counting Ii Statistics Worksheet Page 5