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GATE | GATE-CS-2004 | Question 75

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  • Last Updated : 28 Jun, 2021
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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


Answer: (C)

Explanation: This question is slightly ambiguous. So first let us understand what question is asking. So in a book, we have letters A-Z and each letter is printed twice, so there are 52 letters. Now we have to color each letter, so we need a pair of colors for that, because each letter is printed twice. Also in a pair, both colors can be some. Now condition is that a pair of colors can’t be used more than once.

So suppose Mala has 3 colors : Red, Blue, Green. She can color as follows : (A,A) : (Red,Red), (B,B) : (Blue,Blue), (C,C) : (Green,Green), (D,D) : (Red,Blue), (E,E) : (Red,Green), (F,F) : (Blue,Green).
Now we don’t have more pairs of colors left, we have used all pairs, but could color only 6 letters out of 26. So question is to find minimum no. of colors, so that we could color all 26 letters.

So if Mala has k colors, she can have k pairs of same colors, thus coloring k letters, then kC2 other pairs of colors, thus coloring kC2 more letters.
So total no. of letters colored = k+kC2=k+k(k1)2=k(k+1)2.
So we want k(k+1)226 i.e. k(k+1)52, so k7, so option (C) is correct.

Source: http://www.cse.iitd.ac.in/~mittal/gate/gate_math_2004.html

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