#229 A wireless sensor grid consists of 21

A wireless sensor grid consists of 21 - Math

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Free Chegg Question

A wireless sensor grid consists of 21 x 11 = 231 sensor nodes that are located at points (i,j) in the plane such that i ϵ {0, 1, ... , 20} and j ϵ {0, 1, 2, ... , 10} as shown in Figure 2.1. The sensor node located at the point (0,0) needs to send a message to a node located at (20, 10). The messages are sent to the destination by going from each sensor to a neighboring sensor located above or to the right. That is, we assume that each node located at point (i,j) will only send messages to the nodes located at (i + 1,j) or (i,j + 1). How many different paths do exist for sending the message from the node (0,0) to node (20, 10)?

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Free Chegg Answer

This is a very simple problem of permutation and combination. The language of the problem makes it look tidy. We are supposed to calculate the number of ways of going from (0,0) to (20,10) on a coordinate axis. The only condition is that you can move to the right or to the top at any node.

I have tried to explain using this grid -

Please ignore the scaling errors and the poor drawing. Just focus on the concept.

You can see that the possible moves at any point is only in two direction.

So, to reach (20,10) from (0,0), we need to take 20 horizontal steps and 10 vertical steps. These steps can be taken any how. I have shown below two of the different possible ways to make you understand clearly.

So, I hope now you can understand the problem corectly.

For sensor at (0,0) to send message to the sensor at (20,10), it will have to send message to any 20 horizontal sensors and any 10 vertical sensors.

So, now this is a case of permutation of alike objects (as all the horizontal steps are alike of one kind and also the vertical steps are alike of other kind).

So, basically this is a permutation of 30 objects (steps) out of which 20 are of one kind and 10 are of other kind.

So, the possible number of arrangements is -

Hence, 30045015 different paths exist to send the message from (0,0) to (20,10).