If each participant of a chess tournament plays exactly one game with each of the remaining participants, then 153 games will be played during the tournament. Find the number of participants.  Choose the most appropriate option
A) 15 B) 16 C) 17 D) 18 E) 19
If the number of participants is 3 (say A, B, C) the number of games played will be 2 (A plays against B and C) + 1 (B plays against C) = 3   Using the same logic, if the number of participants is n, the number of games played will be (n-1) + (n - 2) + (n - 3) + ... 3 + 2 + 1   Given that this sum = 153 = 1 + 2 + 3 + ... ( n - 1)   Sum of first m positive integers is given by m(m+1)/2. So sum of first (n-1) positive integers is (n-1)*n/2   153 = (n-1)*n/2   (n-1)*n = 306   17*18= 
<gadget id="calculator">17*18</gadget>
<output>306</output>
306 (We know that 15^2= 225 so the two consecutive numbers must be greater than 15. Also, 20^2 = 400 so the two numbers must be less than 20. The pair of numbers in between 15 and 20 whose product ends with 6 is 17 and 18)   So n = 18   Answer (18)
<result>D</result>
If each participant of a chess tournament plays exactly one game with each of the remaining participants, then 153 games will be played during the tournament. Find the number of participants.