Punish them

The required number of pairs is $23_{C_2}=\dfrac{23\times 22}{2}=\boxed {253}$ .

The formula for finding the possibilities is:

$\frac{(Quantity of items)!}{(Quantity of picked items)!(The Quantity left)!}$

Let's plug this question in this formula:

$\frac{23!}{2!21!}$

= $\frac{22 * 23}{2}$

= 11 * 23

= 253

The problem is solved with ${N \choose K}$ where N is 23(the naughty students) and K(the number of students we need to pick).

${N \choose K}$ = (23 22)/2 = 23 11 = $\boxed{253}$