SAT Mc Graw Hill 2011

350 MCGRAW-HILL’S SAT

Using Venn Diagrams to Keep

Track of Sets

Some counting problems involve “overlapping

sets,” that is, sets that contain elements that

also belong in other sets. In these situations,

Venn diagrams are very helpful for keeping

track of things.

Example:

A class of 29 students sponsored two field trips: one

to a zoo and one to a museum. Every student at-

tended at least one of the field trips, and 10 stu-

dents attended both. If twice as many students

went to the zoo as went to the museum, how many

students went to the zoo?

Set up a Venn diagram of the situation. We repre-

sent the two sets—those who went to the museum

and those who went to the zoo—as two overlapping

circles, because some students went to both. Notice

that there are three regions to consider. We know that

ten students are in the overlapping region, but we

don’t know how many are

in the other two regions,

so let’s use algebra. Let’s

say that xstudents are in

the first region, represent-

ing those who went to the

museum but not to the

zoo. This means that x+

10 students must have

gone to the museum altogether. Since twice as many

students went to the zoo, the total number in the zoo

circle must be 2(x+10) = 2 x+20. Since 10 of these are

already accounted for in the overlapping region, there

must be 2x+ 20 − 10 = 2 x+10 in the third region. So

now the diagram should look like this:

The total number of students is 29, so

(x) +(10) +(2x+10) = 29

Simplify: 3 x+ 20 = 29

Solve: x= 3

So the number of students who went to the zoo is

2(3) + 20 =26.

The Fundamental Counting Principle

Some SAT questions ask you to count things. Some-
times it’s easy enough to just write out the things in a
list and count them by hand. Other times, though,
there will be too many, and it will help to use the Fun-
damental Counting Principle.

Lesson 5: Counting Problems

To use the Fundamental Counting Principle

(FCP), you have to think of the things you’re

counting as coming from a sequence of

choices. The Fundamental Counting Principle

says that the number of ways an event can hap-

pen is equal to the product of the choices that

must be made to “build” the event.

Example:
How many ways can five people be arranged in
a line?

You might consider calling the five people A, B, C, D,
and E,and listing the number of arrangements. After a
while, though, you’ll see that this is going to take a lot
of time, because there are a lot of possibilities. (Not to
mention that it’s really easy to miss some of them.) In-
stead, think of “building” the line with a sequence of
choices: first pick the first person, then pick the second
person, etc. There are five choices to make, so we’ll
have to multiply five numbers. Clearly, there are
five people to choose from for the first person in line.
Once you do this, though, there are only four people left
for the second spot, then three for the third spot, etc. By
the Fundamental Counting Principle, then, the number
of possible arrangements is 5 × 4 × 3 × 2 × 1 =120.

Example:
How many odd integers greater than 500 and less
than 1,000 have an even digit in the tens place?

This seems a lot harder than it is. Again, think of
“building” the numbers in question. All integers
between 500 and 1,000 have three digits, so building
the number involves choosing three digits, so we will
multiply three numbers to get our answer. If each
number is between 500 and 1,000, then there are only
five choices for the first digit: 5, 6, 7, 8, or 9. If the tens
digit must be even, we have five choices again: 2, 4, 6,
8, or 0. If the entire number is odd, then we have five
choices for the last digit as well: 1, 3, 5, 7, or 9.
Therefore, the total number of such integers is
5 × 5 × 5 =125.

To the

Museum

To the

Zoo

Museum

(but not

Zoo)

Zoo

(but not

Museum)

Both

x 10 2 x + 10

MZ