grid ,
Consider the folowingJoe wants to go from to . At any point he can only go one unit up or one unit right
Now Joe can go to in multiple ways. Each path from to is a set of points that have been covered by Joe ( in a particular order like shown below) For eg.
is a path from to
Now the rule is that no two points and in this set of points ( a path ) should satisfy these properties ( both ) and
how many ways are there to go from to ?
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Let > means Joe go right 1 unit, ^ means Joe go up 1 unit. The special condition of this question is as same as Joe cannot walk up twice consecutive. Hence, the answer is equivalent as the ways to arrange 10 > and 7 ^ , such that no 2 consecutive ^ .
First, separate all 7 ^ ,which means put 1 > in two consecutive ^ . Hence, we got the basic arrangement ^>^>^>^>^>^>^ (we already use 6 >, which left 4 >.) Now, we need to arrange the 4 >. Let see some examples, >>^>>^>^>^>^>^>^>, ^>>>^>>^>^>^>^>^>. We can find that we make 7 ^ as 7 dividing line. Hence, the ways of arrangement is equal to ways of arrange 4 > into 8 distinct bins.
Hence, the answer is ( 4 4 + 8 − 1 ) = 3 3 0