Solving this makes fun!

Let $k$ be a positive integer. The organising committee of a tennis tournament is to schedule the matches for $2k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay $1$ coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. This minimum cost is given by :

$\large\ { { k }^{ a }\left( b{ k }^{ c } + k - d \right) }/{ 2 },$

find $a + b + c + d$ , where $a, b, c, d$ are integers.