`8n` players `P_1, P_2, P_3, ,P_(8n)` play a knock out tournament. It is known that all the players are of equal strength. The tournament is held in three rounds where the players are paired at random in each round. If it is given that `P_1` Wins in the third round .Find the probability of `P_2` loses in the second round.

Eight players P_1, P_2, P_3, ...........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 lt j . Assuming that the players are paired at random in each round, what is the probability that the players P_4 , reaches the final ?

2^n players of equal strength are playing a knock out tournament. If they are paired at randomly in all rounds, find out the probability that out of two particular players S_1a n dS_2, exactly one will reach in semi-final (n in N ,ngeq2)dot

In a knockout tournament 2^(n) equally skilled players, S_(1),S_(2),….S_(2n), are participatingl. In each round, players are divided in pair at random and winner from each pair moves in the next round. If S_(2) reaches the semi-final, then the probability that S_(1) wins the tournament is 1/84. The value of n equals _______.

Sixteen players S_1,S_2, ..., S_(16) play in a tournament. They are divided into eight pairs at random. From each pair a winner is decided on the basis of a game played between the two players of the pair. Assume that all the players are of equal strength.Find the probability that the player S_1 is among the eight winners.

Thirty-two players ranked 1 to 32 are playing in a knockout tournament. Assume that in every match between any two players the better ranked player wins, the probability that ranked 1 and ranked 2 players are winner and runner up respectively is p, then the value of [2//p] is, where [.] represents the greatest integer function,_____.

Sixteen players S_(1),S_(2),…,S_(16) play in a tournament. They are divided into eight pairs at random. From each pair a winner is decided on the basis of a game played between the two players decided to the basis of a game played between the two players of the pair. Assume that all the players are of equal strength. (a) Find the prabability that the player S_(1) is among the eight winners. (b) Find the probability that exactly one of the two players S_(1)and S_(2) is among the eight winners.

116 people participated in a knockout tennis tournament. The players are paired up in the first round, the winners of the first round are paired up in the second round, and so on till the final is played between two players. If after any round, there is odd number of players, one player is given a by, i.e. he skips that round and plays the next round with the winners. The total number of matches played in the tournment is

Sixteen players S_(1) , S_(2) , S_(3) ,…, S_(16) play in a tournament. Number of ways in which they can be grouped into eight pairs so that S_(1) and S_(2) are in different groups, is equal to

If the sum of the n terms of a G.P. is (3^(n)-1) , then find the sum of the series whose terms are reciprocal of the given G.P..

Let P(n) denote the statement that n^(2)+n is odd. It is seen that P(n) = P(n + 1). P(n) is true for all.