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Find the number of paths from  to  if each step must be in a right-ward direction. (For example, one possible path is )

 

 

 Apr 28, 2023
 #1
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-1

There are 18 possible paths:

 

A C E G I
A B C E G I
A C D E G I
A C E F G I
A C E G H I
A C E F H I
A B D E F H I
A B C D E G I
A C E F G H I
A B D E F H I
A B D F G H I
A B C D E G I
A B C E F G I
A B D E F G H I
A C D E F G H I
A B C D F G H I
A B C D E F H I
A B C D E F G H I
 

 Apr 28, 2023
 #2
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Notice that the number of ways to get to point C is the sum of the number of ways to get to point A and point B. We can then use this logic on all the other points until we get to point I. 

 

There is 1 way to get to point A and 1 way to get to point B, so there are 2 ways to get to point C. 

 

There are 2 ways to get to point C and 1 way to get to point B so there are 3 ways to get to point D. 

 

There are 3 ways to get to point D and 2 ways to get to point C so there are 5 ways to get to point E. 

 

There are 5 ways to get to point E and 3 ways to get to point D so there are 8 ways to get to point F. 

 

There are 8 ways to get to point F and 5 ways to get to point E so there are 13 ways to get to point G. 

 

There are 13 ways to get to point G and 8 ways to get to point F so there are 21 ways to get to point H. 

 

There are 21 ways to get to point H and 13 ways to get to point G so there are \(\color{brown}\boxed{34}\) ways to get to point D. 

 Apr 28, 2023
 #3
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To find the number of paths from A to I, if each step must be in a right-ward direction, we can use the principle of multiplication. 

Starting from A, we have only one option to move to C. From C, we can move to either D or E. From D, we can only move to E, and from E, we can move to either F or G. From F, we can only move to G, and from G, we can move to either H or I. Finally, from H, we can only move to I.

Therefore, the total number of paths from A to I, moving only in a right-ward direction, is the product of the number of choices at each step:

1 (choice at A) x 2 (choices at C) x 1 (choice at D) x 2 (choices at E) x 1 (choice at F) x 2 (choices at G) x 1 (choice at H) = 4

So there are 4 possible paths from A to I, if each step must be in a right-ward direction.

 Apr 28, 2023
 #4
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We can solve this problem by counting the number of ways to reach each letter on the right. There is only one way to reach A, since it is the first letter. There are two ways to reach B, since it is one step to the right of A. There are three ways to reach C, since it is two steps to the right of A. And so on. Therefore, the total number of paths from A to I is:

1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.

Guest Apr 29, 2023
 #5
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We can solve this problem by counting the number of ways to reach each letter on the right. There is only one way to reach A, since it is the first letter. There are two ways to reach B, since it is one step to the right of A. There are three ways to reach C, since it is two steps to the right of A. And so on. Therefore, the total number of paths from A to I is:

1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.

 Apr 29, 2023
 #6
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There are 5 rightward steps from A to I. We can choose any of the 5 steps to be the first step, and then any of the remaining 4 steps to be the second step, and so on. Therefore, there are 5⋅4⋅3⋅2⋅1=120​ possible paths.

 Apr 29, 2023
 #7
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There are 5 rightward steps to go from A to I. Each step can be made in 4 different ways: A to C, C to D, D to F, F to H, and H to I. Therefore, there are 4^5=1024​ possible paths from A to I.

 Apr 29, 2023

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