= Solution
The first command reads a rectangular data file into `a`; `header=T` treats the first line as variable names. There are twelve observations, identified by student names, and nine binary attributes. A one records presence of an attribute and a zero records absence. This is a description of the encoding, not a claim that every presence is equally important for a scientific application.
`dist(a, metric="binary")` compares rows using <Jaccard distance>. If $n_{11}$ counts shared presences and $n_{10},n_{01}$ count mismatched presences, then
$$
\boxed{d(x,y)=\frac{n_{10}+n_{01}}{n_{11}+n_{10}+n_{01}}.}
$$
Joint absences do not enter the denominator, unlike the <simple matching coefficient> or <normalized Hamming distance>. `dist2full(d)` expands the stored pairwise distances into a symmetric <dissimilarity matrix>; `round(...,2)` changes only its displayed precision. The clustering command still receives the original unrounded distances.
For example, Tom and Martin have six shared presences and only one mismatch, giving $d=1/7=0.142857\ldots$. Taeko and Luitgard have three shared presences and four mismatches, giving $d=4/7=0.571428\ldots$. LingChen and Mohammad share only the meat attribute and differ on four others, giving $d=4/5=0.8$. The diagonal is zero, and the entries are symmetric because exchanging the two observations exchanges $n_{10}$ and $n_{01}$.
`hclust(d, method="compact")` constructs an <agglomerative hierarchical clustering> object by <complete-linkage clustering>. Its distance between two current groups is
$$
d(A,B)=\max_{x\in A,\,y\in B}d(x,y).
$$
Start with singleton groups, merge a pair with smallest intergroup distance, update these maximum cross-pair distances, and repeat until one group remains. The object stores the merge structure, merge heights and a leaf order suitable for a <dendrogram>; assigning it to `h` does not itself display a graph. The historical https://sites.oxy.edu/lengyel/m150/Sueselbeck/helpfiles/hclust.html[S-Plus help] identifies `compact` with complete linkage and describes the plotting object.
The first merge is Tom with Martin at $1/7$. At the next height $1/5$, either LinYee with HuiChin or HuiChin with Meg may merge: the distances are tied. The following <dendrogram> shows one valid resolution, using original row order to resolve ties. In this tree, LinYee joins HuiChin at $1/5$, Pio joins Meg at $1/4$, and Nicolas joins Tom–Martin at $3/8$. The latter height is $\max\{d(\text{Nicolas},\text{Tom}),d(\text{Nicolas},\text{Martin})\}=\max\{3/8,2/7\}$, illustrating the complete-linkage update.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-43-clustering.png]
{title=Complete-linkage dendrogram and Jaccard-distance heatmap for the binary student profiles, with one valid tie convention}
For the displayed tree, a cut at height $0.65$ gives three groups: Taeko–Luitgard; Alet–Tom–Martin–Nicolas–Mohammad; and LinYee–Pio–LingChen–HuiChin–Meg. The last two merges occur at $5/7$ and $4/5$. \b[The hierarchy is a description under the chosen binary distance and linkage rule; it neither supplies an intrinsically correct number of groups nor guarantees a unique tree in the presence of ties.] Changing the treatment of joint absences, or the weighting of attributes, can change that description.
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