This bank machine has four rows of buttons
1 2 3
4 5 6
7 8 9
0
Sam has a 4-digit bank number that uses one button from each row. For example his bank number could be 0571.
How many different possibilities for Sam's bank number are there?
The answer should be 648
3 choices on each row except the last, so
3*3*3*1
That gives the number of digit choices. Now, for each choice of 4 digits, there are 4! ways to arrange them, so the final tally is
3*3*3*1*4!
To find the number of different possibilities for Sam's bank number, we need to determine how many choices he has for each digit.
For the first digit, Sam can choose any of the 10 buttons (0-9) since there are no restrictions.
For the second digit, since Sam has already chosen one button from the first row, he has 3 options remaining in that row (1, 2, and 3). Similarly, he has 3 options in the second row, 3 options in the third row, and 1 option in the fourth row. Therefore, Sam has a total of 3 x 3 x 3 x 1 = 27 options for the second digit.
Following the same logic, for the third digit, Sam will have 2 options remaining in each row. Therefore, he has a total of 2 x 2 x 2 x 1 = 8 options for the third digit.
Finally, for the fourth digit, Sam will have 1 option remaining in each row. Therefore, he has a total of 1 x 1 x 1 x 1 = 1 option for the fourth digit.
To find the total number of possibilities for Sam's bank number, we multiply the number of options for each digit together: 10 x 27 x 8 x 1 = 2,160.
Therefore, there are 2,160 different possibilities for Sam's bank number.
To find the number of possibilities for Sam's bank number, we need to consider the number of choices for each digit and then multiply them together.
Since there are 4 rows and each digit of the bank number uses one button from each row, we have the following choices:
1. For the first digit, Sam can choose any of the 4 rows, so there are 4 choices.
2. For the second, third, and fourth digits, Sam can choose any of the 3 remaining rows each time. So for each of these digits, there are 3 choices.
To find the total number of possibilities, we multiply the number of choices for each digit together:
4 choices for the first digit × 3 choices for the second digit × 3 choices for the third digit × 3 choices for the fourth digit = 4 × 3 × 3 × 3 = 108
However, this calculation includes cases where Sam chooses the same row for multiple digits. Since the bank number must use one button from each row, these cases are not valid.
To exclude these invalid cases, we need to consider that Sam must choose a different row for each digit.
For the first digit, there are 4 choices. Once Sam has made this choice, there are only 3 rows left to choose from for each of the next three digits, so there are 3 choices for each of these digits.
To find the total number of valid possibilities, we multiply the number of choices for each digit together:
4 choices for the first digit × 3 choices for the second digit × 3 choices for the third digit × 3 choices for the fourth digit = 4 × 3 × 3 × 3 = 108
Therefore, there are 108 different possibilities for Sam's bank number.