Answer to Question #152981 in Discrete Mathematics for Miry

Question

A group of 21 students participates in a discrete mathematics competition. There are Q questions that have to be answered. For each question exactly 4 students are assigned to work together, while the others take a short break. The questions are distributed to ensure that every pair of students only works together on exactly one exercise. This works out exactly!

How many exercises does the competition have?

(Hint: Consider how many pairs occur in a group of 4 students.)

Answer 1

$\text{the number of different sets of length 2 of 4 elements is}$

$\frac{4*3}{6}=6$

$\text{we have Q questions or Q sets of length 4}$

$\text{which is Q*6 different sets of length 2}$

$\text{by convention, each pair of students works together }$

$\text{on only one exercise}$

$\text {that is, the number of distinct subsets of length 2 from}$

$\text{a group of students is Q*6}$

$\text{on the other hand:}$

$\frac{21*20}{2}=210$ $\text{number of different pairs from the whole}$

$\text{group of students}$

$Q*6 = 210;Q=35$

Answer:Q=35 - number of questions