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65 votes

Mala has the 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?

  1. $9$
  2. $8$
  3. $7$
  4. $6$
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9 Answers

2 votes
2 votes
I was doing K(K-1)>=26

but i understood that this is wrong because=>BECAUSE k(k-1) would give only the distinct pairs.  and not the repeated pairs.
1 votes
1 votes
K² should be at least 52 to color all colors.
=> K >= 7
Therefore minimum 7 colors are required.
0 votes
0 votes

Let say we just have 6 English letters, A, B, C, D, E, F  and k = 3(Assuming I have red, green, black color) with 3c2 = we get just 3 combinations, i.e p1(red, green), p2(black, green) and p3(red, black) with these three pairs we can assign to 3 pair of letters  p1(AA) , p2(BB), p3(CC) but wait what about other 3 more pairs CC, EE, FF they can be assigned 3 colored as follows(Red Red ) , (Green Green) and (Black Black),   hence we can use same approach. 

For our problem we have 26pairs and if k = 6 we can just have 6c2 = 15 can be colored and 6 more with same color for pair so this is invalid we could not color all 26 pairs.

with k = 7, we can color 7C2 = 21 and plus 7 more so we have covered all 26 pairs.

Therefore C is the answer

0 votes
0 votes

 

LETTERS COLOURS(Ci)
A A C1 C1
B B C2 C2
C C C1 C2
D D C3 C3
E E C1 C3
F F C2 C3
G G C4 C4
H H C1 C4
I I C2 C4
J J C3 C4
K K C5 C5
L L C1 C5  
M M C2 C5
N N C3 C5
O O C4 C5
P P C6 C6
Q Q C1 C6
R R C2 C6
S S C3 C6
T T C4 C6
U U C5 C6
V V C7 C7
W W C1 C7  
X X C2 C7
Y Y C3 C7
Z Z C4 C7

=> Minimum 7 colours are needed.

Answer:

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