Question

Mala has a colouring book in which each English letter is drawn two times. She wants to paint each of these 52 prints with one of k colours, such that the colour-pairs used to colour any two letters are diffrent. Both prints of a letter can also be coloured with the same colour. What is the minimum value of k that satiesfies this requirment ?

• A. 8
• B. 9
• C. 6
• D. 7

Answer 1

The correct option is C 6

The problem reduces to finding how many distinct orderd colour pairs $(C_1,C_2)$ are possible with k colors.
Since the first color $C_1$ can be anyone of the k colours and the second color $C_2$ also can be any one of the k colors (both prints of a letter can be colored with same color), the total no. of such order color pairs is equal to $k\times k=k^2$.
Since each pair of letters must be colored with diffrent color pairs, at least 26 color pairs are required to do this.
Therefore the requirments is $k^2\ge 26$
The minimum value of k that satisfies this equation is k = 6.