If you are looking for the number of ways to place those pieces on the first two ranks of the board, there are:
16 places to put the king
15 places left to put the queen
C(14,2) = (14*13)/2 ways to pick 2 of the remaining 14 places for the two bishops (*)
C(12,2) = (12*11)/2 ways to pick 2 of the remaining 12 places for the two knights
C(10,2) = (10*9)/2 ways to pick 2 of the remaining 10 places for the two rooks.
1 way to fill the remaining 8 spaces with pawns
Multiply all those counts together to get the total. My calculator says that is 64,864,800 different arrangements are possible.
That allows any piece or pawn to be placed on any of the 16 squares. In particular, the (*) note on the bishops is because this placement allows the two bishops to be on the same colored square.
Chess variants that people actually play usually place the major pieces all on the 1st rank with pawns filling the 2nd rank, as in standard chess. Also, one bishop on each color is usually enforced. Fischer Random Chess, invented by the great Bobby Fischer, requires that, and also (for special castling rules) that the king is on a square between the two rooks. With those rules, there are:
4 places to put one bishop on a black square
4 places to put the other bishop on a white square
C(6,3) = (6*5*4)/(1*2*3) = 20 ways to pick 3 of the remaining 6 to put the king between 2 rooks
3 places to put the queen on one of the 3 remaining squares
1 way to put the knights on the remaining two squares
Multiply those and you get 960 starting positions for one side, and another name for this game is "Chess960".