In a knockout tournament, `2^(n)` equally skilld players, `S_(1), S_(2), …, S_(2^(n))` are participating. In each round, players are divided in pairs at random and winner from each pair moves to 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 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

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 ?

Let S_(n) be the sum of first n terms of an A.P. If S_(2n) = 5s_(n) , then find the value of S_(3n) : S_(2n) .

Explain the stereochemistry of S_(N)2 and S_(N)1 reaction with suitable examples.

Let S_n denote the sum of first n terms of an A.P. If S_(2n)=3S_n , then find the ratio S_(3n)//S_ndot

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

Let S_n denote the sum of first n terms of the sequence 1,5,14,30,55,..... then prove that S_n-S_(n-1)=sumn^2 .

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

Thirty two players ranked 1 to 32 are playing is 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 (A) 16/31 (B) 1/2 (C) 17/31 (D) none of these

Let S_1 , S_2, ,,,, be squares such that for each n ge 1 the length of a side of S_n equals the length of a diagonal of S_(n +1) . If the length of a sides of S_1 is 10 cm, then for which of the following values of n in the ares of S_n less than 1 sq. cm ?