Final answer:
To find out how many people fall into exactly two of the three categories, we apply the inclusion-exclusion principle and calculate that 186 members of club XYZ fall under exactly two categories.
Step-by-step explanation:
The problem describes a scenario with three overlapping categories of club members: post-graduates, sportsmen, and politicians, along with given percentages for different intersection groups. To determine how many people come under exactly any two of the three categories, we use a principle in set theory that involves subtracting the intersection of all three categories from each two-category intersection. This principle is sometimes referred to as the inclusion-exclusion principle.
First, we'll add the percentages of members that fall into exactly two of the three categories: post-graduates and sportsmen (12%), sportsmen and politicians (13%), and post-graduates and politicians (14%). But we must remember that we've included the members who fall into all three categories (4%) three times, once in each percentage, so we must subtract it twice to account for this: (12% + 13% + 14%) - (2 × 4%) = 39% - 8% = 31%.
Next, we will find the total percentage of members involved in any of the categories. This is done by adding up the individual category percentages and subtracting the overlapping percentages (including the triple overlap once): 42% + 43% + 44% - (12% + 13% + 14%) + 4% = 129% - 39% + 4% = 94%. Those who are neither in the given categories make up the remaining percentage to complete 100%. As there are 36 members who are neither, and they represent 6% of the total (100% - 94%), the total membership can be calculated as 36 members ÷ 6% = 600 members.
To find the exact number of members that fall into any two categories, we calculate 31% of 600 members, which gives us: 0.31 × 600 = 186 members.