Questions: Use a Tree Diagram to Perform a Multiple-Part Counting Task Example Find the number of three-letter combinations that can be written using only the letters A, B, and C, assuming that a. repeated letters are allowed

Use a Tree Diagram to Perform a Multiple-Part Counting Task

Example Find the number of three-letter combinations that can be written using only the letters A, B, and C, assuming that a. repeated letters are allowed

Transcript text: Use a Tree Diagram to Perform a Multiple-Part Counting Task Example Find the number of three-letter combinations that can be written using only the letters $A, B$, and $C$, assuming that a. repeated letters are allowed

Solution

Solution Steps

To solve the problem of finding the number of three-letter combinations using the letters A, B, and C with repeated letters allowed, we can use the concept of permutations with repetition. Since each position in the three-letter combination can be filled by any of the three letters, and there are three positions, the total number of combinations is calculated by raising the number of choices (3) to the power of the number of positions (3).

Step 1: Determine the Number of Letters and Positions

We have a total of $ 3 $ letters: $ A, B, C $. We need to form combinations of $ 3 $ letters.

Step 2: Calculate the Total Combinations

Since repeated letters are allowed, each of the $ 3 $ positions in the combination can be filled by any of the $ 3 $ letters. Therefore, the total number of combinations can be calculated using the formula:

\[ \text{Total Combinations} = \text{Number of Letters}^{\text{Number of Positions}} = 3^3 \]

Step 3: Perform the Calculation

Calculating $ 3^3 $:

\[ 3^3 = 27 \]