Mala has a colouring book in which each English letter is drawn two times. She…
2004
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 different. Both prints of a letter can also be coloured with the same colour. What is the minimum value of k that satisfies this requirement ?
- A.
9
- B.
8
- C.
7
- D.
6
Attempted by 2 students.
Show answer & explanation
Correct answer: C
Each English letter needs a distinct colour-pair for its two printed copies. Since the two copies of a letter are not ordered, and both copies may have the same colour, the number of possible colour-pairs from k colours is the number of combinations with repetition of size 2: C(k + 1, 2) = k(k + 1)/2. We need k(k + 1)/2 >= 26. For k = 6, this gives 21, which is not enough. For k = 7, this gives 28, which is enough. Therefore, the minimum value of k is 7.