Allie and Betty want to know how many 3-letter combinations, such as BMW or DDT, are possible. (Letters are allowed to repeat, as in DDT or BOB.) Allie thinks there can be 26+26+26 three-letter combinations, whereas Betty thinks the number is 26*26*26. Which girl, if ether, is right, and why? Explain your answers clearly and thoroughly, drawing on the meaning of multiplication.
there are 26 choices for each letter. So, multiply.
To determine who is correct, let's consider the possible number of options for each position in a three-letter combination.
Allie's Approach:
Allie thinks that there are 26 + 26 + 26 combinations for each position. Let's break down this approach:
- For the first position, there are 26 options (A-Z).
- For the second position, there are 26 options again (A-Z).
- Similarly, for the third position, there are 26 options (A-Z).
Therefore, using Allie's approach, the total number of combinations would be 26 + 26 + 26 = 78.
Betty's Approach:
Betty thinks that there are 26 * 26 * 26 combinations for each position. Let's analyze this approach:
- For the first position, there are 26 options (A-Z).
- For the second position, there are again 26 options (A-Z).
- Similarly, for the third position, there are 26 options (A-Z).
According to Betty's multiplication approach, the total number of combinations would be 26 * 26 * 26 = 17,576.
Conclusion:
Betty's approach is correct. The reason is that for each position, there are 26 options (the 26 letters of the English alphabet), and since each position is independent of others, we multiply the number of options for each position.
Using the multiplication principle, the total number of combinations is obtained by multiplying the number of options for each position: 26 * 26 * 26 = 17,576.
Therefore, Betty's answer of 17,576 is the correct number of possible three-letter combinations.