There are five hobby clubs in a college —photography, yachting, chess, electronics and gardening. The gardening group meets every second day, the electronics group meets every third day, the chess group meets every fourth day, the yachting group meets every fifth day and the photography group meets every sixth day. How many times do all the five groups meet on the same day within 180 days?

Correct Option (d)

To determine the number of times all five hobby clubs meet on the same day, it is necessary to identify the Least Common Multiple (LCM) of their individual meeting frequencies. The clubs meet at the following intervals:

• Gardening group: every 2nd day
• Electronics group: every 3rd day
• Chess group: every 4th day
• Yachting group: every 5th day
• Photography group: every 6th day

The meeting frequencies are 2, 3, 4, 5, and 6 days. The LCM of these numbers represents the smallest number of days after which all clubs will meet simultaneously again.

The calculation for the LCM is as follows:

$$\text{LCM}(2,3,4,5,6)=2^2\times 3\times 5=60$$

This result indicates that all five clubs will meet on the same day every 60 days.

To find the total number of such simultaneous meetings within a 180-day period, the total period is divided by the LCM:

Number of meetings = 180 days / 60 days/meeting = 3 times.

Therefore, all five groups meet on the same day 3 times within the specified 180-day duration.

Incorrect Options:

Options (a) 5, (b) 18, and (c) 10 are incorrect. These values do not correspond to the mathematically derived frequency of simultaneous meetings. The unique interval for all clubs to meet concurrently is 60 days, as determined by the Least Common Multiple of their individual meeting cycles. Any deviation from this calculation or its application over the 180-day period would lead to an erroneous number of meetings.