There are 90 students in a classroom. How many relationships are there in this classroom? Show how you got this.
Hint: Consider this 2 students in the class would be 1 relationship, whereas 7 students in the class would yield 21 relationships.
What I think: I'm super stuck on this. I think you use factorials but how?
combinations of 2 at a time?
for example for 7 students
n!/[ 2! (7-2)! ]
= 7 /{2*5!}
= 7 * 6 /2 = 21 well that checks so
90! /[ 2!(88!) ]
= 90 * 89/2
= 45*89
= 4005
= 7! /{2*5!}
= 7 * 6 /2 = 21 well that checks so
To determine the number of relationships in the classroom, we need to consider that a relationship exists between any two students in the class.
Let's break down the problem:
- With 2 students, we have 1 relationship.
- With 3 students, we have 3 relationships (1 relationship between student 1 and student 2, 1 relationship between student 1 and student 3, and 1 relationship between student 2 and student 3).
- With 4 students, we have 6 relationships (1+2+3 relationships).
- With 5 students, we have 10 relationships (1+2+3+4 relationships).
It seems like the number of relationships is increasing according to the triangular number sequence (1, 3, 6, 10, etc.), which can be obtained by summing consecutive positive integers.
To determine the number of relationships with 90 students, we can use the formula for the triangular number sequence:
Triangular number = (n * (n + 1)) / 2,
where n represents the number of students.
Substituting n = 90 into the formula, we get:
(90 * (90 + 1)) / 2 = (90 * 91) / 2 = 4095.
Hence, there are 4095 relationships in the classroom with 90 students.