Problem

Source: Greece Archimedes Junior 2020

Tags: combinatorics



We are having  99 equal circles in a row and in the interior, we write inside them all the numbers from 1 up to 99 (one number in each circle).We color each of the circles with one of the two colors available: red and green. A coloring is good if it has the ability: Red circles lying in the interval of the numbers from 1 up to 50  are more than the red circles lying in the interval of the numbers from 51  up to 99 . a) Find how many different colorings can be constructed. b) Find how many different good colorings can be constructed. (Note: Two colorings are different, if they have different color in at least one of their circles.)