17 views

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 ay 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. $6$

edited ago | 17 views

total number of alphanumeric letter = $26$
Each is printed twice the no. of letters = $26×2$ = $52$
case 1: If Mala has $k$ colors, she can have k pairs of same colors.(red,red),(blue,blue)...etc
case 2: She also can have $kc_{2}$ different pairs in which each pair is having different colors.(blue,green),(red,green)...etc

k colors let pairs (a,b) can be choose with k color and b can be choose with k-1 color=$k*(k-1)$

and (a,b) (b,a) is same thing we divide by 2

$k*(k-1)/2$=$kc_{2}$

So  minimum no. of colors, so that we could color all 26 letters.=$case 1+case 2\geqslant26$
$k+kc_{2} ≥ 26$
$k+k(k-1)/2 ≥ 26$
$k(k+1)/2 ≥ 26$
$k(k+1) ≥ 52$
$k(k+1) ≥ 7*8$
$k≥7$

ago by Active
edited ago