Eight players P 1 , P 2 , . . . . . . . . . . . . . P 8 play a knock – out tournament It is known that whenever the players P i and P j play, the player P i will win if i 4 reaches the final?

Question

Aditi Chauhan · Answer

Hello Student,

Please find the answer to your question

The number of ways in which P₁, P₂. . . . . . . . . . . . . . . . . . . , P₈ can be paired in four pairs.

= 1/4! x ⁸C₂ x ⁶C₂ x ⁸C₂ x ⁴C₂ x ²C₂ = 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 play against each other in between P₁, P₂, P₃ and remaining player will play against one of the players from P₅, P₆, P₇, P₈ and P₄ plays against one of the remaining three from P₅, P₆, P₇, P₈

This can be possible in ³ C₂ x ^{4 C1} x ³C₁ = 36 ways

∴ Prob. that P₄ and exactly one of P₅ . . . . . . . . . . . P₈ reach second round

= 36/105 = 12/35

If P₁, P_i, P₄ and P_j where i = 2 or 3 and j = 5 or 6 or 7 reach the second round then they can be paired in 2 pairs in

1/2! x ⁴ C₂ x ² C₂ = 3 ways

But P₄ will reach the final if P₁ plays against P_i and P₄ plays against P_j

Hence the prob. that P₄ reach the final round from the second = 1/3.

∴ prob. that P₄ reach the final is 12/35 x 1/3 = 4/35.

Thanks
Aditi Chauhan
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Eight players P 1 , P 2 , . . . . . . . . . . . . . P 8 play a knock – out tournament It is known that whenever the players P i and P j play, the player P i will win if i 4 reaches the final?

Eight players P₁, P₂, . . . . . . . . . . . . . P₈ play a knock – out tournament It is known that whenever the players P_i and P_j play, the player P_i will win if i ₄ reaches the final?

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Eight players P 1 , P 2 , . . . . . . . . . . . . . P 8 play a knock – out tournament It is known that whenever the players P i and P j play, the player P i will win if i 4 reaches the final?

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 4 reaches the final?

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Eight players P₁, P₂, . . . . . . . . . . . . . P₈ play a knock – out tournament It is known that whenever the players P_i and P_j play, the player P_i will win if i ₄ reaches the final?