Question

In a 7-by-7 checkerboard, as shown,

two unit squares will be chosen at
random and without replacement. What
is the probability that the two squares
are adjacent to each other (share a side)?
Express your answer as a common
fraction.

Answer 1

Explanation:

Mia is saving money each week. Mia's savings are represented by the graph below. Find the rate of change.

0

Weeks

Total Savings (Dollars)

x

y

0

Weeks

Total Savings (Dollars)

Mia's Savings

(0,$6)

(10,$131)

Answer

Attempt 1 out of 2

dollar sign$

per week per week

Answer 2

Let's calculate the total number of ways to choose two squares without replacement on a 7-by-7 checkerboard. There are a total of

7

×

7

=

49

7×7=49 unit squares.

For the first square, there are 49 choices. For the second square, there are 48 choices (since one square has already been chosen). So, there are

49

×

48

49×48 ways to choose two squares.

Now, let's calculate the number of ways to choose two squares that are adjacent to each other (share a side). For each square, there are 4 adjacent squares (excluding the squares on the edges and corners, which have fewer adjacent squares). So, for the first square, there are 4 choices, and for the second square, there are 3 choices.

Therefore, there are

4

×

3

4×3 ways to choose two adjacent squares.

The probability that the two squares are adjacent is the number of ways to choose two adjacent squares divided by the total number of ways to choose two squares:

�

(

adjacent

)

=

4

×

3

49

×

48

P(adjacent)=

49×48

4×3

​

Now, simplify the fraction:

�

(

adjacent

)

=

12

2352

P(adjacent)=

2352

12

​

So, the probability that the two squares chosen are adjacent to each other is

12

2352

2352

12

​

, which can be further simplified if needed.