Question 1201865
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The following 7 × 22 grid is divided into squares that are 1 unit by 1 unit.
The shortest possible path on this grid from A to B is 29 units long. 
One such path is shown in the figure. Let X be the set of all 29-unit-long paths from A to B.
Compute |X|, the number of 29-unit-long paths from A to B.
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Indeed,  the figure is not shown,  and it can perplex a reader.


But notice that  29 = 7 + 22,  and it makes the meaning of the problem  UNIQUE  even without a plotted figure.


The points  A  and  B  are opposite vertices of the  7x22 rectangle 
and the shown path is a path on the grid,  comprised of vertical and horizontal segments 
of the grid from corner  A  to corner  B,  such that a path represents 
a piece-wide monotonic function.


For better understanding, this grid has 7 segments in one direction and 22 segments in other direction;
but the number of the vertices of the grid is (7+1) = 8 in one direction and (22+1) = 23 in the other direction.


We can think that  22  segments of the grid are horizontal and  7  segments are vertical,
and A is the left lowest corner of the grid,  while B is the upper right corner.
So,  we can think that the associated function is a piece-wide monotonically increasing.


The meaning of the problems remains the same at any other disposition of the grid
and the corners A and B.


Notice that vertical increments of the function are not necessary one vertical segment 
of the grid:  it can be two segments or three etc.,  but the governing rule is that the 
associated function is a piece-wide  MONOTONIC - it provides the shortest path.


Then it becomes  CLEAR  that this problem is the same as to ask,
in how many ways  22  (undistinguishable)  objects can be distributed (placed) in 8 
distinguishable boxes in a way that the empty boxes are allowed.


In other words,  it can be re-formulated in this way: how many solutions this equation


            {{{x[1]}}} + {{{x[2]}}} + {{{x[3]}}} + {{{x[4]}}} + {{{x[5]}}} + {{{x[6]}}} + {{{x[7]}}} + {{{x[8]}}} = 22           (1)


may have in integer non-negative numbers    {{{x[i]}}},  i = 1, 2, 3, . . . , 8.


<pre>
In this form, it is a typical problem to be solved by the "stars-and-bars" method.


About this method, see this Wikipedia article 

https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29


or my lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =https://www.algebra.com/algebra/homework/Permutations/Stars-and-bars-method-for-Combinatorics-problems-2.lesson>Stars and bars method for Combinatorics problems</A> 

in this site.



The answer is THIS: the number of solutions to equation (1) is  

    {{{(22+8-1)!/(22!*(8-1)!)}}} = {{{29!/(22!*7!)}}} = {{{(29*28*27*26*25*24*23)/(1*2*3*4*5*6*7)}}} = 1560780.


<U>ANSWER</U>.  The number of paths in this problem is  1560780.
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Solved.