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The Brushstrokes Art Academy is organizing a special project where students and teachers collaborate on paintings. The academy has 50 participants in total, including both students and teachers. Each student will create one painting with every other student and one painting with each teacher.
When two students collaborate, they create a small canvas painting.
When a student collaborates with a teacher, they create a large canvas painting.
At the end of the project, there are exactly 141 large canvas paintings completed.
How many small canvas paintings (student-to-student collaborations) were created during this project?
Step 1: Define the Variables
Let x be the number of teachers.
Let $(50 – x)$ be the number of students.
Step 2: The Collaboration Equation
Each teacher paints with each student. If the total is 141 large canvases:
Step 3: Solve the Quadratic
Unlike the messy original numbers, this factors perfectly! We need two numbers that multiply to 141 and add to -50. Those numbers are -3 and -47.
So, x = 3 or x = 47. Since a school usually has more students than teachers, we can confidently say there are 3 Teachers and 47 Students.
Step 4: The Handshake Problem (Combinatorics)
To find the number of small canvases (student-to-student collaborations), we need to find how many unique pairs can be made from 47 students. This is a classic combinations formula as seen in the diagram.