How many routes can a ball take as it travels from A to G, from A to H, from A to I, from A to J, and from A to K?

**Technifi Expert’s Answer:**

Consider a *Galton board* as shown below:

The number in each hexagon represents the total number of different routes that a ball from A can take to reach the top of that particular hexagon.

**Understand the Problem** **: **A ball dropped from A on striking the vertex of an hexagon has equal probablility of going to either left or right. So the ball can have many different routes to reach a paticular hexagon.

**Devise a Plan** **: **When the ball touche a vertex, it can either go left or right. To find the total number of possible routes between two point, concentrate on the initial and final point and proceed by first considering motion in one particular direction, say left, this will give one route. Now, the other route will be the one with all the initial motion in left direction and final motion towards right. Then consider the next possibility where the second last motion is towards right and so on. Write the number of routes in each hexagon and then look for a pattern.

From the diagram considered above, a pattern can be observed that, all the outer hexagons have 1 inside them. So, Hexagon B and F will have value 1. Also note that all hexagons other than the outer ones have value equal to the sum of values of two hexagons on the above row sharing boundary with the hexagon under consideration. Hence hexagon C has value, hexagon D will have value , hexagon E will have value .

**Carry Out the Plan** **: **Suppose there is a hexagon placed at G, H, I, J, K and L. Then using the pattern discovered, one can find that, hexagon G has value 1, hexagon H will have value 8, hexagon I will have value 28, hexagon J will have value 56, hexagon K will have value 70 and hexagon L will have value 56. Hence, the *Galton Board* can be filled as shown below:

The value in hexagon is the total number of diferent routes that the ball can take to reach the top of that particular hexagon, but here at the indicated position instead of hexagons there are ‘bins’, but still the value found will give the number of different routes. Therefore, to reach point G from A there is 1 roue, to reach H from A there are 8 routes, to reach I from A there are 28 routes, to reach J from A there are 56 routes, to reach K from A there are 70 routes and to reach L from A there are 56 routes.

**Review the Solution** **: **One can draw out each route and verfiy that the total number of reported routes is indeed the correct number.

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