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To determine the number of ways to choose 3 pastries from 8, we use the combination formula:
For the spreads, since we are selecting 1 out of 5, there are:
The total number of combinations is the product of these:
We need to count 6-digit numbers where each digit is either 3 or 4, and no two adjacent digits are the same.
Start with the first digit: it can be either 3 or 4 — 2 choices.
For each subsequent digit, it must differ from the previous one, so there is exactly 1 choice for each next digit (the opposite of the prior digit).
Thus, the number of such sequences is:
Wait — this is incorrect for 6 digits. Actually, after choosing the first digit (2 choices), each next digit is uniquely determined (only one option to alternate), so:
Indeed, the only two valid sequences are:
Count how many numbers fall into each class:
