The given line graph shows the record of the number of tickets sold at the theatre counter on the first $6$ days of a show. On how many days was the number of tickets sold less than that sold on the previous day?

Let's list the number of tickets sold on each day from Day $1$ to Day $6$:
* Day 1: $1000$
* Day 2: $1500$ (Increased from Day $1$)
* Day 3: $2500$ (Increased from Day $2$)
* Day 4: $2000$ (Decreased from Day $3$, since $2000<2500$) -> Count = 1
* Day 5: $3000$ (Increased from Day $4$)
* Day 6: $2500$ (Decreased from Day $5$, since $2500<3000$) -> Count = 2

Thus, there are exactly $2$ days (Day $4$ and Day $6$) on which the sales dropped compared to the previous day.