Problem
1982 AHSME Problem 25
The adjacent map is part of a city: the small rectangles are rocks, and the paths in between are streets. Each morning, a student walks from intersection A to intersection B, always walking along streets shown, and always going east or south. For variety, at each intersection where he has a choice, he chooses with probability \frac{1}{2} whether to go east or south. Find the probability that through any given morning, he goes through C.
![[asy] defaultpen(linewidth(0.7)+fontsize(8)); size(250); path p=origin--(5,0)--(5,3)--(0,3)--cycle; path q=(5,19)--(6,19)--(6,20)--(5,20)--cycle; int i,j; for(i=0; i<5; i=i+1) { for(j=0; j<6; j=j+1) { draw(shift(6*i, 4*j)*p); }} clip((4,2)--(25,2)--(25,21)--(4,21)--cycle); fill(q^^shift(18,-16)*q^^shift(18,-12)*q, black); label("A", (6,19), SE); label("B", (23,4), NW); label("C", (23,8), NW); draw((26,11.5)--(30,11.5), Arrows(5)); draw((28,9.5)--(28,13.5), Arrows(5)); label("N", (28,13.5), N); label("W", (26,11.5), W); label("E", (30,11.5), E); label("S", (28,9.5), S);[/asy]](/static/aops/ffd6c029970fcec169cae363da55912394de8155.png)
\text{(A)}\frac{11}{32}\qquad \text{(B)}\frac{1}{2}\qquad \text{(C)}\frac{4}{7}\qquad \text{(D)}\frac{21}{32}\qquad \text{(E)}\frac{3}{4}
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