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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 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?

- $9$
- $8$
- $7$
- $6$

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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.=$case 1+case 2\geqslant26$

$k+kc_{2} ≥ 26$

$k+k(k-1)/2 ≥ 26$

$k(k+1)/2 ≥ 26$

$k(k+1) ≥ 52$

$k(k+1) ≥ 7*8$

$k≥7$

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