Question

In a computer science department, a student club can be formed with either 10 members from first year or 8 members from second year or 6 from third year or 4 from final year. What is the minimum no. of students we have to choose randomly from department to ensure that a student club is formed?

Answer 1

Pigeonhole principle : Let $q_1, q_2, . . . , q_n$  be positive integers.

If $q_1+ q_2+ . . . + q_n – n + 1$ objects are put into $n$ boxes, then either the 1st box contains at least $q_1$ objects, or the 2nd box contains at least $q_2$  objects, . . ., the nth box contains at least $q_n$  objects.

Given $q_1=10, q_2=8, q_3=6, q_4=4.$

The number of boxes = the number of different student courses:members from first year, members from second year, members from third year and members from last year n=4.

Therefore the minimum number of students required to ensure department club to be formed is

$$10+8+6+4-4+1=25$$