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Hi, I have a few questions and was wondering if anyone could explain them to me

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Hi, I have a few questions and was wondering if anyone could explain them to me

In the following grid, there are  \(\binom{14}{9}\)

paths of length 14 from A to B, where each step goes right or up. Find the number of these paths that pass through edge NO.

In the following grid, there are \(\binom{14}{9}\) paths of length 14 from A to B, where each step goes right or up. Find the number of these paths that pass through edge PQ.

How many arrangements of the numbers  are there where the sum of any two adjacent numbers is odd?

Mar 6, 2019

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Not expert on this....but....

First one....we are only interested in the number of paths to "N"   [ from here, there is only one way to get to O ]

Note that  the movements that get us to  N from A  are ( right, up, up, up, up, up)

And in this set  we can choose any 1 of the 6 positions for "right" = C(6, 1)....or...alternatively, any 5 of the 6 positions for "up".....so....the number of paths to N  are  C(6,1) = C(6,5)  =  6  paths

Second one....again, we are only interested in getting to P

We have the following moves from A to P  =

(R, R, R, R, R, U, U, U, U )

We can choose any 5 of 9 positions for R  or, alternatively, any 4 of the 9 positions for U

So....the total paths are C(9, 5) = C(9,4)  = 126 paths

Mar 6, 2019