An office has a total of 95 employees. Each employee participates in one or more of three activities, i.e. music, dance and skit, during the annual festival. The number of employees participating in only music is 20, whereas 14 participate in only skit, and 17 participate in only dance. A total of 8 employees participate in both music and skit only, 12 employees participate in all three activities, and 15 employees participate in both dance and skit only, whereas 9 employees participate in both music and dance only. Find the total number of employees participating in skit?

Question

Prepp Team · Accepted Answer

Understanding Employee Participation in Activities

The question describes a scenario in an office with 95 employees, where each employee participates in at least one of three activities: music, dance, and skit. We are given the number of employees participating in specific combinations of these activities and asked to find the total number of employees participating in skit.

Analyzing the Given Information

We are provided with detailed numbers for participation in various categories. Let M represent music, D represent dance, and S represent skit.

Only music: 20 employees
Only dance: 17 employees
Only skit: 14 employees
Music and skit only: 8 employees (participate in M and S, but not D)
Dance and skit only: 15 employees (participate in D and S, but not M)
Music and dance only: 9 employees (participate in M and D, but not S)
All three (music, dance, and skit): 12 employees

These numbers represent disjoint regions in a Venn diagram illustrating the participation in the three activities. The sum of employees in all these disjoint regions should equal the total number of employees, as everyone participates in at least one activity.

Let's verify the total:

Total employees = (Only M) + (Only D) + (Only S) + (M & D only) + (M & S only) + (D & S only) + (M & D & S)

$20 + 17 + 14 + 9 + 8 + 15 + 12 = 95$

The sum matches the total number of employees given, confirming that all employees are accounted for in these categories.

Calculating Total Participants in Skit

The question asks for the total number of employees participating in skit. This includes everyone who participates in skit, regardless of whether they also participate in music or dance or both. In terms of the disjoint regions identified above, the total number of employees participating in skit is the sum of employees in all regions that are part of the 'Skit' circle in a Venn diagram.

These regions are:

Those who participate in only skit (S only)
Those who participate in music and skit only (M & S only)
Those who participate in dance and skit only (D & S only)
Those who participate in all three activities (M & D & S)

Using the given numbers, we sum the employees in these categories to find the total participating in skit:

Total employees participating in Skit = (S only) + (M & S only) + (D & S only) + (M & D & S)

Total employees participating in Skit = $14 + 8 + 15 + 12$

Let's perform the addition:

$14 + 8 = 22$

$22 + 15 = 37$

$37 + 12 = 49$

Therefore, the total number of employees participating in skit is 49.

Summary of Skit Participants

Participation Category	Number of Employees
Only Skit	14
Music and Skit only	8
Dance and Skit only	15
Music, Dance, and Skit (All three)	12
Total Skit Participants	49

The total number of employees participating in skit is 49.

Revision Table: Key Concepts Revisited

Concept	Description	Relevance to Problem
Set Theory	Branch of mathematics dealing with sets (collections of objects).	This problem uses concepts like sets (activities), subsets (combinations of activities), and intersections.
Venn Diagram	A visual representation of sets and their relationships, showing overlaps (intersections).	Helps visualize the different groups of employees and how their participation overlaps.
Disjoint Sets/Regions	Sets or regions with no elements in common (e.g., 'only music' and 'only dance').	The problem provides numbers for specific disjoint regions within the Venn diagram.
Union of Sets	The combination of all elements in two or more sets.	The total number of employees participating in skit is the union of several disjoint groups within the 'Skit' set.
Intersection of Sets	Elements common to two or more sets (e.g., employees in both music and skit).	Categories like 'music and skit only' or 'all three' represent specific intersections.

Additional Information: Understanding "Only" vs. "And" in Set Problems

It's crucial to understand the difference between phrases like "music and skit only" and "music and skit".

"Music and Skit only" refers to employees who participate in music and skit, but not dance. This is a specific, exclusive group. In a three-set Venn diagram, this is the region where the Music and Skit circles overlap, but *outside* the Dance circle.
"Music and Skit" (without "only" or "but not...") refers to all employees who participate in both music and skit. This group includes those who do music and skit only, *as well as* those who participate in music, skit, and dance (all three). This is the entire overlapping region of the Music and Skit circles.

In this problem, the numbers provided are for the disjoint "only" categories (only M, only D, only S, M&D only, M&S only, D&S only) and the "all three" category. This makes calculating the total for a single activity straightforward by summing the relevant disjoint regions within that activity's set.