There is only one red ball, but there are two black balls. So there are two methods: red ball+1 black ball, red ball +2 black ball. C (subscript 2, superscript 1) is not taken from the red ball. Then multiply it by 2, because this is an arrangement, not a combination. C (subscript 3, superscript 1)*C (subscript 3, superscript 1) is also arranged.

You misunderstood the latter part. C (subscript 3, superscript 1) stands for choosing one of the red ball and the black ball. The process should be as follows:

P(X= 1| Z=0)=P(X= 1，Z=0)/P(Z=0)

P(X= 1, Z=0)=C (subscript 2, superscript 1)*2/C (subscript 6, superscript 1)*C (subscript 6, superscript 1).

P(Z=0)=C (subscript 3, superscript 1)*C (subscript 3, superscript 1)/C (subscript 6, superscript 1)*C (subscript 6, superscript 1).

So the answer came out.