An ant begins its path at $A$, travels only right or down, and remains on the line segments shown. The number of different paths from $A$ to $C$ that pass through $B$ is
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-011.jpg?height=246&width=431&top_left_y=2029&top_left_x=1302)
We begin by labelling additional points, as shown.

Beginning at $A$, the ant may travel right (to $D$ ) or down (to $F$ ).

Assume the ant begins by travelling right, to $D$. From $D$, if the ant continues travelling right (to $E$ ) the path cannot pass through $B$ (since the ant can travel only right or down).

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-055.jpg?height=263&width=409&top_left_y=476&top_left_x=1563)
From $D$, the ant must travel down to $B$. From $B$, there are two paths that end at $C$, one travelling right to $G$ and then down to $C$, and another travelling down to $I$ and then right to $C$. Therefore, there are two paths in which the ant begins by travelling right: $A-D-B-G-C$ and $A-D-B-I-C$.

Assume the ant begins by travelling down, to $F$. From $F$, if the ant continues travelling down (to $H$ ) the path cannot pass through $B$ (since the ant can travel only right or down).

Thus, from $F$, the ant must travel right to $B$. From $B$, there are two paths that end at $C$, as discussed above.

Therefore, there are two paths in which the ant begins by travelling down: $A-F-B-G-C$ and $A-F-B-I-C$.

There are 4 different paths from $A$ to $C$ that pass through $B$.

ANSWER: $(\mathrm{C})$