<p>There are 5 historical monuments, 6 gardens, and 7 shopping malls in a city. In how many ways a tourist can visit the city if he visits at least one shopping mall?</p>
Step-by-Step Solution
Key Concept: The tourist must visit at least one shopping mall, so we count all possible visiting combinations for monuments and gardens (each can be visited or not), then multiply by the number of non-empty subsets of shopping malls.
<p><strong>Step 1:</strong> For 5 historical monuments: each monument can be either visited or not visited → 2^5 ways (including visiting none).</p><p><strong>Step 2:</strong> For 6 gardens: each garden can be either visited or not visited → 2^6 ways (including visiting none).</p><p><strong>Step 3:</strong> For 7 shopping malls: the tourist must visit at least one mall. Total subsets of 7 malls = 2^7, but exclude the empty set (visiting no malls) → (2^7 - 1) ways.</p><p><strong>Step 4:</strong> By the multiplication principle, total ways = 2^5 × 2^6 × (2^7 - 1).</p><p>∴ Answer: <strong>2^5 × 2^6 × (2^7 - 1)</strong> = 32 × 64 × 127 = <strong>260,096</strong></p>
Correct Answer: 2^5 × 2^6 × (2^7 - 1)