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) ? )

(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) ? )