Holetaug

2022-07-07

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) ? )

alomjabpdl0

Each color appears in at most 20 drawings, and there are 12 colors, so there are at most $12\cdot 20=240$ 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).

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