Answer
$56$ ways
Work Step by Step
There are 8 swimmers and the top 3 advance to the next round.
Order does not matter here since so solve for $_{8}$C$_{3}$.using the formula $_nC_r=\dfrac{n!}{(r!(n-r)!)}$:
$_8C_3=\dfrac{8!}{(3!(8-3)!)}$
$_8C_3=\dfrac{8!}{3!(5!)}$
$_8C_3=\dfrac{8*7*6*5*4*3*2*1}{(3*2*1)(5*4*3*2*1)}$
Simplify to get $C(8,3)=56$.