Answer: (iii) $R = 3^d - 2^{d+1} + 1$

Derivation

For a dataset with $d$ distinct items, every possible association rule is formed by:

1. Choosing a non-empty itemset $Y$ from the $d$ items to be the "antecedent ∪ consequent" (the full rule itemset).
2. Splitting $Y$ into a non-empty antecedent (X) and non-empty consequent (Y − X).

For each item, it can go into: antecedent, consequent, or neither (not part of the rule) → giving $3^d$ total combinations.

From this $3^d$, we must subtract:

• The cases where no item is in the consequent (i.e., only antecedent+neither combos): $2^d$
• The cases where no item is in the antecedent (only consequent+neither combos): $2^d$
• Add back the case counted twice (no item in either — the empty rule): $+1$

$$R = 3^d - 2 \cdot 2^d + 1 = 3^d - 2^{d+1} + 1$$

This is the standard result from Tan/Steinbach/Kumar's Introduction to Data Mining — confirming option (iii).