Question: What is the minimum number of students required in a discrete mathematics class to be
sure that at least six will receive the same grade, if there are five possible grades, $A, B,
C, D$, and $F$?
Solution: The minimum number of students to ensure that at least six students
receive the same grade is the smallest integer $N$ such that
$⌈N/6⌉ = 6$.
The smallest such integer is
$N = 5 \cdot 5 + 1 = 26$.
If you have only $25$ students, it is possible for there to be five who have received each
grade so that no six students have received the same grade. Thus, $26$ is the minimum
number of students needed to ensure that at least six students will receive the same grade.
Doubt: Why we get $5.5+1$?
Can someone clearly explain?
I think because of $K+1$ on Pigeon Hole Principle, if so why $5$ multiply by $5$ again?
Best Answer
Let's say we have $26=5\cdot 5+1$ students but suppose that no $6$ students get the same grade. Then at for each grade there are at most $5$ students getting that particular grade. But then that means there are at most $5\cdot 5$ students all together, which is a contradiction.