It has been established that, upon the addition of multivalent cations, long DNA chains in an aqueous solution exhibit a remarkable discrete transition from a coil state to a compact state at the level of a single chain. In this study, we investigated the polyelectrolyte nature of DNA with the experimental methodology of single-DNA observation, and provide a theoretical interpretation. We examined the effects of co-ions with different valencies (Cl(-), SO4(2-), PO4(3-)) on DNA compaction. As a result, we found that co-ions with a greater valency induce the coil state rather than the compact state. Based on a simple model with mean-field approximation that considered ion pairing, we show how the increase in entropy of small ions contributes to the stability of the compact state, by overcoming entropic penalties such as elastic confinement of the chain and a decrease in the translational freedom of counterions accompanied by charge neutralization.