To solve this problem, we need to apply hierarchical clustering with complete linkage to cluster the data points into four clusters.

Steps in hierarchical clustering with complete linkage:

1. Calculate the distance matrix: Compute the Euclidean distance between each pair of points.
2. Start with each point as its own cluster.
3. Merge clusters: In each step, we merge the two clusters that have the smallest maximum pairwise distance between any of their members. This is the key feature of complete linkage.
4. Continue merging until there are four clusters.

Step 1: Compute the distance matrix.

Let's calculate the Euclidean distance between each pair of points. The points are:

• A(1, 1)
• B(1, 2)
• C(2, 3)
• D(3, 1)
• E(3, 7)
• F(2, 5)
• G(2, 6)

The Euclidean distance formula is:

$d(p_1, p_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

	A	B	C	D	E	F	G
A	0	1	2.24	2.0	6.32	4.47	5.39
B	1	0	1.41	2.83	5.83	3.61	4.47
C	2.24	1.41	0	2.83	4.47	2.0	3.0
D	2.0	2.83	2.83	0	6.0	4.47	5.39
E	6.32	5.83	4.47	6.0	0	2.0	1.0
F	4.47	3.61	2.0	4.47	2.0	0	1.0
G	5.39	4.47	3.0	5.39	1.0	1.0	0

Step 2: Clustering

Now let's apply the complete linkage method step by step:

1. Initial clusters: {A}, {B}, {C}, {D}, {E}, {F}, {G}.
2. First merge: {E} and {G} (smallest distance = 1.0). Clusters: {A}, {B}, {C}, {D}, {E,G}, {F}.
3. Second merge: {E,G} and {F} (distance = 1.0). Clusters: {A}, {B}, {C}, {D}, {E,F,G}.
4. Third merge: {A} and {B} (distance = 1.0). Clusters: {A,B}, {C}, {D}, {E,F,G}.
5. Fourth merge: No more merges as we now have 4 clusters.

Final clusters:

• {A,B}
• {C}
• {D}
• {E,F,G}

Conclusion:

The correct clustering is option (a): {A,B}, {C}, {D}, {E,F,G}.

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Would you like further details or have any questions?

Here are 5 related questions you might find helpful:

1. How does complete linkage differ from single linkage in hierarchical clustering?
2. Can hierarchical clustering always guarantee optimal clusters for all datasets?
3. How would the clusters change if we used single linkage instead of complete linkage?
4. What are some common use cases for hierarchical clustering in real-world scenarios?
5. How does the dendrogram represent hierarchical clustering?

Tip: In hierarchical clustering, the choice of linkage (complete, single, or average) can drastically change the resulting clusters, so it’s important to consider the nature of your data before choosing the method.