We need to take 6 steps to reach the final state. 1st,2nd,3rd,4th,5th,6th.
Out of these 6 steps 3 are horizontal step and 3 are vertical step.
It is our choice when we will take the vertial steps. Assume we have taken 2nd,4th,and 6th step vertial.
Selecting these 3 steps out of 6 steps has = $\binom{6}{3} = 20$ different ways.
Once we have selected, when to go vertical, there is no choice of permutation between a,b and c.
Among vertical steps we must follow order $a\rightarrow b\rightarrow c$ only. No other way.
Additionally once we fixed the vertical steps , horizontal step positions are also got fixed and they are also in the order $x\rightarrow y\rightarrow z$
=> Total no of accepted strings = 20. (D NONE)
