下の図のように，$xy$平面上に，$x$軸に平行な道，$y$軸に平行な道，直線$y=-x$に平行な道があるものとする．これらの道を通って，原点Oから点A$(4,\ 4)$まで行くとき，以下の各場合に道順の総数を求めよ． \setlength\unitlength{1truecm} \begin{center} \begin{picture}(7,7)(0,0) \put(2.5,0.5){\line(1,0){3}} \put(0.5,2.5){\line(1,0){6}} \put(0.5,3.5){\line(1,0){6}} \put(0.5,4.5){\line(1,0){6}} \put(0.5,5.5){\line(1,0){5}} \put(1.5,6.5){\line(1,0){4}} \put(0.5,2.5){\line(0,1){3}} \put(2.5,0.5){\line(0,1){6}} \put(3.5,0.5){\line(0,1){6}} \put(4.5,0.5){\line(0,1){6}} \put(5.5,0.5){\line(0,1){6}} \put(6.5,1.5){\line(0,1){3}} \put(6.5,4.5){\line(-1,1){2}} \put(6.5,3.5){\line(-1,1){3}} \put(6.5,2.5){\line(-1,1){4}} \put(6.5,1.5){\line(-1,1){5}} \put(5.5,1.5){\line(-1,1){4}} \put(5.5,0.5){\line(-1,1){5}} \put(4.5,0.5){\line(-1,1){4}} \put(3.5,0.5){\line(-1,1){3}} \put(2.5,0.5){\line(-1,1){2}} \multiput(0.5,1.5)(0,0.20){5}{\line(0,1){0.1}} \multiput(1.5,0.5)(0.2,0){5}{\line(1,0){0.1}} \put(0,1.5){\vector(1,0){7}} \put(1.5,0){\vector(0,1){7}} \put(0.3,1.1){$-1$} \put(0.9,0.4){$-1$} \put(5.6,5.5){A} \put(3.6,3.9){B} \put(0.1,3.4){C} \put(1.1,1.1){O} \put(6.9,1.1){$x$} \put(1.1,6.9){$y$} \put(2.2,1.1){1} \put(3.2,1.1){2} \put(4.2,1.1){3} \put(5.2,1.1){4} \put(6.4,1.1){5} \put(1.2,2.1){1} \put(1.2,3.1){2} \put(1.2,4.1){3} \put(1.2,5.1){4} \put(1.2,6.3){5} \put(5.5,5.5){\circle*{0.15}} \put(3.5,4){\circle*{0.15}} \put(0.5,3.5){\circle*{0.15}} \put(1.5,1.5){\circle*{0.15}} \end{picture} \end{center}
(1) 最短経路で行く場合．
(2) 点B$(2,\ 2.5)$を通らずに，最短経路で行く場合．
(3) 点C$(-1,\ 2)$を通り，道のりが$8+\sqrt{2}$になる場合．
(4) 道のりが$8+\sqrt{2}$になる場合．
(5) $0 \leqq x \leqq 4,\ 0 \leqq y \leqq 4$の部分だけを通り，道のりが$8+\sqrt{2}$になる場合．