Five college hobby clubs meet periodically: gardening every 2 days, electronics every 3 days, chess every 4 days, yachting every 5 days, and photography every 6 days. Within a span of 180 days, how many times do all five clubs meet on the same day (excluding the starting day)?

Introduction:
When periodic events coincide, the interval between coincidences equals the least common multiple of their individual periods. We apply that idea to five meeting schedules measured in days.

Given Data / Assumptions:

• Periods: 2, 3, 4, 5, and 6 days
• Window: 180 days
• We count coincidences after day 0, excluding the starting day meeting, which is a standard counting convention for such questions.

Concept / Approach:
Find LCM(2, 3, 4, 5, 6). The groups meet together every LCM days. The number of such meetings within 180 days, excluding the start, is 180 / LCM.

Step-by-Step Solution:

Verification / Alternative check:
The common meeting days fall on day 60, day 120, and day 180 when we exclude day 0. Counting day 0 would give 4, but the problem asks within 180 days excluding the starting day by our stated convention.

Why Other Options Are Wrong:
5, 10, and 18 overcount; 4 would include the starting day, which the present interpretation excludes.

Common Pitfalls:
Including the starting day when the phrase within 180 days is interpreted as after the start. Always clarify the counting convention.

Final Answer:
3