P₁,P₂ ,.....P₈ an eight players participating in a tournament. If i < j, then P_i will win the match against P_j. Players are paired up randomly for first round and winners of this round again paired up for the second round and so on. The probability that P₄ reaches in the final is $\frac{4}{\mathrm{K}}$then ‘K’ equal to ___

Let A₁ be the event that in the first round the four winners are ${\mathrm{P}}_1,{\mathrm{P}}_4,{\mathrm{P}}_{\mathrm{i}},{\mathrm{P}}_{\mathrm{j}}$ where i∈{2 , 3 j∈{5,6,7}and let A₂ be the event that out of the four winners in the first round, P₁ and P₄ reaches in the final. The event A₁  will occur, if P₄ plays with any of  (say with P₆ ) and  P₁ ,P₂ and P₃ are not paired with  P₅ , P₇ and  P₈ . Further A₂  will occur if  P₁ plays with P_i .
The required probability $=\frac{\mathrm{P}\left({\mathrm{A}}_1\cap {\mathrm{A}}_2\right)}{\mathrm{P}\left({\mathrm{A}}_1\right)}$
 

(Here we have used the concept of division into groups).