Classical models built from the math you already know
How many groups are actually in the data? k-means never answers that on its own: it groups points into k clusters once k is already decided, but where that number comes from is left entirely up to you. Pick k too small and two genuinely different groups get squashed into one. Pick it too large and one honest group gets sliced into pieces that were never really separate. The elbow method gives an honest way to choose k instead of guessing.
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▶ How Many Clusters? Elbow & Silhouette