Imagine seeking a path from a given source to given destination in a Manhattan-like city grid. How many path there are to arrive to point a to point b?

This variant assumes that that the grid is a 3D space

ANSWER:

(define manhattan-3d

 (lambda (i j k)
  (cond
   ((and (= j 0) (= k 0)) 1)
   ((and (= j 0) (= i 0)) 1)
   ((and (= k 0) (= i 0)) 1)
   ((= i 0) (+ (manhattan-3d i (- j 1) k)
          (manhattan-3d i j (- k 1))))
   ((= k 0) (+ (manhattan-3d i (- j 1) k)
          (manhattan-3d (- i 1) j k)))
   ((= j 0) (+ (manhattan-3d i j (- k 1))
          (manhattan-3d (- i 1) j k)))
   (else
      (+ (manhattan-3d i (- j 1) k)
     (manhattan-3d (- i 1) j k)
     (manhattan-3d i j (- k 1))))
  )
 )
)

(manhattan-3d 0 0 7) ;1 (manhattan-3d 2 0 2) ;6 (manhattan-3d 1 1 1) ;6 (manhattan-3d 1 1 5) ;42 (manhattan-3d 2 3 1) ;60 (manhattan-3d 2 3 3) ;560