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Question

Eight players P1,P2,....P8 play a knock out tournament. It is known that whenever the players Pi and Pj play, the player Pi will win if i<j. Assuming that the players are paired at random in each round, what is the probability that the player P4 reaches the final ?

A
435
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B
535
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C
635
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D
135
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Solution

The correct option is A 435
The number of ways in which P1,P2,...P8 can be paired in four pairs
14!(8C2)(6C2)(4C2)(3C2)=14!×8!2!6!×6!2!4!×4!2!2!×1
=14!×8×72!×1×6×52!×1×4×32!×1=8×7×6×52.2.2.2=105
Now, at least two players certainly reach the second round in between P1,P2 and P3. And P4 can reach in final if exactly two players against each other in between P1,P2,P3 and remaining players will play against one of the players from P5,P6,P7,P8 and P4 play against one the remaining three from P5...P8.
This can be possible in 3C2×4C1×3C1=3.4.3.=36 ways
Therefore probability that P4 and exactly one of P5...P8 reach second round =36105=1235
If P1,Pi,P4 and Pj where i=2,3 and j=5 or 6 or 7 reach the second round, then they can be paired in 2 pairs in 12!(4C2)(2C2)=3 ways.
But P4 will reach final round from the second =13
Therefore probability the P4 reach the final is =1235×13=435

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