A letter is taken out at random from ‘ASSISTANT’ and another is taken out from ‘STATISTICS’. The probability, that they are same letters is $\frac{K}{90}$ then $K =__$

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answer is 19.

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Detailed Solution

ASSISTANT ………AA I N SSS TT
STATISTICS ……….A II C SSS TTT
Here N and C are common. Same letters can be A, I, S, T
Probability of choosing A is $\frac{2 C_{1}}{9 C_{1}} \times \frac{1}{10 C_{1}} = \frac{1}{45}$
Probability of choosing I is $\frac{1}{9 C_{1}} \times \frac{2 C_{1}}{10 C_{1}} = \frac{1}{45}$
Probability of choosing S is $\frac{3 C_{1}}{9 C_{1}} \times \frac{3 C_{1}}{10 C_{1}} = \frac{1}{10}$
Probability of choosing T is $\frac{2 C_{1}}{9 C_{1}} \times \frac{3 C_{1}}{10 C_{1}} = \frac{1}{15}$
$∴$ Required Probability $= \frac{1}{45} + \frac{1}{45} + \frac{1}{10} + \frac{1}{15} = \frac{19}{90}$