Inclusion-exclusion principle (?) in a counting problem A group of pre-school children is drawing pictures ( one child is making one picture ) using 12-colours pencil set. Given that
(i) each pupil employed 5 or more different colours to make his drawing;
(ii) there was no identical combination of colours in the different drawings;
(iii) the same colour appeared in no more than 20 drawings,
find the maximum number of children who have taken part in this drawing activity.
( As each child can be identified with his/her unique combination of colours, the number of children can not exceed 
C(5,12) + C(6,12) + C(7,12) +...+ C(12,12)
But how to NARROW it using the condition (iii) ? )

HINT: Each color appears in at most $20$ drawings, and there are $12$ colors, so there are at most $12\cdot20=240$ drawings. However, each drawing uses at least $5$ colors, so ... ? See if you can finish it from here; I’ve left the conclusion in the spoiler-protected block below.

 So each drawing is counted at least $5$ times, and there are therefore at most $\frac{240}5=48$ drawings (and hence at most $48$ children).

