No they didn't. That's the point. That's why they have no mandate.The conservatives won a majority, and formed a coalition. If LibDems crashed out, the Tories still have a mandate as they won the majority. The majority of the electorate voted for the Tories, therefore they have a mandate. I didn't say the Coalition, as that's as you said, a by-product. It's not like the Tories wanted it all that, they just wanted enough seats to out vote labour safely.
Edit: actually, an overall majority of votes cast is extremely rare in this country. Even a 40% share of the vote in this country is rare. But in this instance, the Tories didn't even win the majority of seats, which is all that is needed for a mandate to govern in this country.