Correct Answer - Option 2 : 17

Let there be x players

1st player will play with x - 1 players twice = 2(x- 1 ) matches.

2nd player will play with x-2 players as with 1st player he has already played = 2( x-2) matches

So, total number of matches = 2(x -1) + 2(x- 2) + 2(x-3) + .......+2(2) + 2(1) + 2(0)

= 2 ( 1 + 2 + ..... + (x-3) + (x-2) + (x -1) )

= 2 (x-1)x/2

= (x-1)x

⇒x (x-1) = 272

⇒x (x-1) = 17 × 16

⇒x = 17

So, the number of players participated = x = **17**

Hence, **option 2** is the correct answer.