This problem is an application of the stars and bars method with additional restrictions. Here, we are selecting a dozen (12) donuts from 6 different varieties, and the total number of maple donuts must be between 3 and 7.

Step 1: Define the variables

Let the 6 varieties of donuts be represented as $x_1, x_2, \dots, x_6$, where:

• $x_1$ represents the number of maple donuts,
• $x_2, x_3, x_4, x_5, x_6$ represent the number of the other 5 types of donuts.

We need to find the number of ways to choose the donuts such that the total number of donuts is 12, and the number of maple donuts $x_1$ satisfies $3 \leq x_1 \leq 7$.

Step 2: Break the problem into cases

We will consider the number of maple donuts $x_1$ for each case, and then count how many ways we can distribute the remaining donuts among the other varieties.

For each case, the total number of donuts $x_2 + x_3 + x_4 + x_5 + x_6$ is $12 - x_1$.

Case 1: $x_1 = 3$

If $x_1 = 3$, then the remaining 9 donuts need to be distributed among the 5 other varieties. This is a classic stars and bars problem, where the number of ways to distribute 9 indistinguishable donuts into 5 distinguishable varieties is: $\text{Number of ways} = \binom{9+5-1}{5-1} = \binom{13}{4}$

Case 2: $x_1 = 4$

If $x_1 = 4$, then the remaining 8 donuts need to be distributed among the 5 other varieties. The number of ways to distribute 8 donuts is: $\text{Number of ways} = \binom{8+5-1}{5-1} = \binom{12}{4}$

Case 3: $x_1 = 5$

If $x_1 = 5$, then the remaining 7 donuts need to be distributed among the 5 other varieties. The number of ways to distribute 7 donuts is: $\text{Number of ways} = \binom{7+5-1}{5-1} = \binom{11}{4}$

Case 4: $x_1 = 6$

If $x_1 = 6$, then the remaining 6 donuts need to be distributed among the 5 other varieties. The number of ways to distribute 6 donuts is: $\text{Number of ways} = \binom{6+5-1}{5-1} = \binom{10}{4}$

Case 5: $x_1 = 7$

If $x_1 = 7$, then the remaining 5 donuts need to be distributed among the 5 other varieties. The number of ways to distribute 5 donuts is: $\text{Number of ways} = \binom{5+5-1}{5-1} = \binom{9}{4}$

Step 3: Calculate the combinations

Now we calculate the binomial coefficients:

• $\binom{13}{4} = 715$
• $\binom{12}{4} = 495$
• $\binom{11}{4} = 330$
• $\binom{10}{4} = 210$
• $\binom{9}{4} = 126$

Step 4: Add the results

The total number of ways is the sum of the possibilities for each case: $715 + 495 + 330 + 210 + 126 = 1876$

Thus, the total number of ways to select one dozen donuts with between 3 and 7 maple donuts is $\boxed{1876}$.

Would you like more details or have any questions?

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Here are 5 related questions to consider:

1. How does the stars and bars method work in more detail?
2. What if the number of maple donuts could be any number between 0 and 12?
3. How would the problem change if the store offered more than 6 varieties of donuts?
4. How can generating functions be used to solve problems like this one?
5. What if the selection was made without replacement of donuts?

Tip: When working with combinatorics, it's helpful to break problems into manageable cases and apply well-known formulas like stars and bars for counting possibilities efficiently.