A mini license plate for a toy car must consist of a number followed by two letters. Each letter must be a C, A or R. Each number must be a 3 or 7. Repetition of letters is permitted.

Use the counting principle to determine the number of points in the sample space.
Construct a tree diagram to represent this situation.List the sample space.
Determine the exact probability of creating a mini license plate with a C. Give solution exactly in reduced fraction form.

2 choices for the number, followed by 3x3 choices for the letters

number of plates = 2x3x3 = 18

the second part is somewhat ambigious.
Can both spots be C ? I will assume that

1. one C = 2x1x2 + 2x2x1 = 8
2 two C;s = 2x1x1 = 2

number of cases = 10

prob of a C = 10/18 = 5/9

SO CONFUSED STILL....

3CC

3CA
3CR
3AA
3AC
3AR
3RR
3RA
3RC , that's 9 now do the ones starting with 7 for another 9
for a total of 18

now count how many of those 18 have a C in them .
Including the column starting with 7, I count 10 with a C in them

What part is still confusing?

okay...i get it now thank you Reiny :)

To answer these questions, we need to follow a step-by-step process:

1. Use the Counting Principle:
The Counting Principle states that if there are m ways of doing one thing and n ways of doing another thing after it, then there are m x n ways of doing both things together.

In this case, there are 2 choices for the number (3 or 7), and 3 choices for each of the two letters (C, A, or R). Therefore, by using the counting principle, we can determine that there are:

2 (choices for the number) x 3 (choices for the first letter) x 3 (choices for the second letter) = 18 points in the sample space.

2. Construct a Tree Diagram:
To represent the situation, we can construct a tree diagram. Start by drawing two branches from the main point for the number (3 and 7). From each number branch, draw three branches for each letter choice (C, A, and R). The resulting tree diagram will have 18 endpoints representing all combinations of numbers and letters.

Number
/ \
3 7
/ \ / \
C A R C A R

3. List the Sample Space:
Based on the tree diagram, we can list all the possible combinations:
3CC, 3CA, 3CR, 3AC, 3AA, 3AR, 3RC, 3RA, 3RR
7CC, 7CA, 7CR, 7AC, 7AA, 7AR, 7RC, 7RA, 7RR

4. Determine the Exact Probability of Creating a Mini License Plate with a C:
We are looking for the probability of getting a C as the first letter in the mini license plate. We found that there are 18 possible combinations in the sample space.
Out of those 18 combinations, 6 start with the letter C (3CC, 3CA, 3CR, 7CC, 7CA, 7CR).

Therefore, the probability of creating a mini license plate with a C is 6/18. We can simplify this fraction to get the exact probability in reduced form:
6/18 = 1/3.

So, the exact probability of creating a mini license plate with a C is 1/3.