Sat math Essentials

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P R O B L E M S O LV I N G
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1 6 7

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To solve combination questions, use this formula:
n
C
r
n
r
P
!
r
(
n
n
!
r
)!
r
!
C
number of combinations
n
the number of items 
r
number of items in each permutation
For example, to determine the number of three-letter combinations from a group of seven letters 
(
ABCDEFGH
), use the following values:
n
7 and 
r
3.
Plug in the values into the formula:
7
C
3
(
n
n
!
r
)!
r
!
(7
7!
3)!3!
4
7
!3
!
!
7
3
6
2
5
1
21
6
0
35
Therefore there are 35 three-letter combinations from a group of seven letters.
Practice Question 
A film club has five memberships available. There are 12 people who would like to join the club. How many
combinations of the 12 people could fill the five memberships?
a.
60
b.
63
c.
792
d.
19,008
e.
95,040
Answer
c.
The order of the people doesn’t matter in this problem, so it is a combination question, not a permuta-
tion question. Therefore we can use the formula 
n
C
r
(
n
n
!
r
)!
r
!
, where 
n
the number of people who
want the membership 
12 and 
r
the number of memberships 
5.
n
C
r
(
n
n
!
r
)!
r
!
12
C
5
(12
12!
5)!5!
12
C
5
7
1
!
2
5
!
!
12
C
5
12
C
5
12
C
5
95
1
,
2
0
0
40
12
C
5
792
Therefore, there are 792 different combinations of 12 people to fill five memberships.
12 
11 
10 
9 
8
5 
4 
3 
2 
1
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1
(7 
6 
5 
4 
3 
2 
1)5!
7 
6 
5 
4 
3 
2 
1
(4 
3 
2 
1)(3!)
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P R O B L E M S O LV I N G
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1 6 8

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P r o b a b i l i t y
Probability
measures the likelihood that a specific event will occur. Probabilities are expressed as fractions. To find
the probability of a specific outcome, use this formula:
Probability of an event 
Example
If a hat contains nine white buttons, five green buttons, and three black buttons, what is the probability of select-
ing a green button without looking?
Probability 
Probability 
Probability 
9
5
5
3
Probability 
1
5
7
Therefore, the probability of selecting a green button without looking is 
1
5
7
.
Practice Question 
A box of DVDs contains 13 comedies, four action movies, and 15 thrillers. If Brett selects a DVD from the
box without looking, what is the probability he will pick a comedy?