## Task 4 - 340514

Balls can fall into the container from opening $A$. On their way down, they are deflected either to the left or to the right each time they reach the upper corner of an obstacle.

a) How many such paths from $A$ to $B$ are there in total?

b) How many such paths from $B$ to $C$ are there in total?

c) How many such paths from $A$ via $B$ to $C$ are there in total?

d) How many such paths from $A$ to $C$ are there in total?

Explain for at least one of the subtasks a), b), c), d) how you found the total number of possible paths!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0144.jpg?height=451&width=448&top_left_y=1048&top_left_x=1341)
For the description of the path of a ball, the deflection to the left or right is indicated by l or r, respectively.

a) On every path from $A$ to $B$, one encounters three forks in the road. To reach $B$, one must turn left exactly twice and right exactly once. Therefore, there are exactly the 3 paths llr, lrl, rll.

b) To get from $B$ to $C$, one must turn left exactly once and right exactly twice. (In each of these three cases, it is either a decision at a fork in the road or a continuation forced by the edge of the container.)

Thus, there are exactly the 3 paths lrr, rlr, rrl.

c) To get from $A$ via $B$ to $C$, after each of the three paths from a), one can follow any of the three paths from b). Therefore, there are a total of $3 \cdot 3=9$ such paths.

d) To get from $A$ to $C$, one must turn left exactly three times and right exactly three times (either at a fork in the road or forced by the edge of the container). Thus, there are exactly the paths

beginning with lll: lllrrr,

beginning with llr: llrlrr, llrrlr, llrrrl, beginning with lrr: lrllrr, lrlrlr, lrlrrl, beginning with lrr: lrrllr, lrrlrl, lrrrll, beginning with rll: rlllrr, rllrlr, rllrrl, beginning with rlr: rlrllr, rlrlrl, rlrrll, beginning with rrl: rrlllr, rrllrl, rrlrll, beginning with rrr: rrrlll, which are a total of 20 paths.

Solutions from the 1st Round 1994 adopted from [5]

### 2.35.2 2nd Round 1994, Class 5