Black has one eye at the circled point. makes miai of the capture at a and the connection underneath at b, capturing two stones.
If White captures the outside stones, Black makes a second eye with . White can play neither at a nor at b because of shortage of liberties. (This gives White slightly more points than Variaion 1. White may also consider which wall needs more eyes, the upper side or the right side.)
The vulgar atari of is a painful mistake. It's a so-called anti-suji, missing completely the proper line of play.
White answers at and the connection underneath at a doesn't work any more. White must not play anti-suji with at b, for Black will descend to c and make the connection work again.
Isn't it also possible to add a before going on with b (solution)? It would threaten to live with c...
grolich: Although the problem is "black to live", the correct solution will be the best way to live, not just ANY way to live. If black threatens to live with a in the diagram white would just capture the corner stone. This also naturally captures the two black stones below, as no more tricks remain, and black is forced to live. white captured 3 stones instead of black capturing two in one line or one in another. That's quite a big difference. Big loss on black's part.
noobish?: Furthermore, the best solution should preserve ko threats. So for example, starting with any of the throw-ins on the top line and then continuing with the correct sequence should not be considered an alternate solution. In general, any sequence beginning with unnecessary forcing moves is inferior to the more direct approach.