Final answer:
To calculate the number of 4-digit PINs where digits 7 and 8 must be beside each other, treat the pair as a single unit, leading to 2 × 6 × 10 × 9 = 1080 possible PINs, but this is not reflected in the provided answer choices.
Step-by-step explanation:
The question asks how many different 4-digit PINs can be created if the digits 7 and 8 must be beside each other. To solve this, we can treat the pair of digits 7 and 8 as one single unit. Since there are two ways to arrange these two digits (78 and 87), we have two variations of this single unit. After grouping these two digits together, we have three 'units' to arrange: the 78 (or 87) unit, plus the two remaining digits. This can be done in 3! = 3 × 2 × 1 = 6 ways. As there are 10 possible digits (0-9) for each of the two remaining positions, we must account for 10 options for the first digit and 9 options for the second digit (since we can't reuse numbers in a PIN). Thus, we have 10 × 9 choices for the other two digits. Combining these, we get the total number of PINs as 2 (ways to arrange 7 and 8) × 6 (ways to arrange the three units) × 10 × 9 (choices for the other two digits). The calculation is 2 × 6 × 10 × 9 = 1080 possible PINs. However, none of the provided answer choices exactly match this number, indicating a possible error in the question or the answer choices.