PS: Harmony

Question

A nice one

giddi77 · Answer

Is it 20?

Please see attached working. To find the number of ways to get to a node we need to add the number of ways to get to the immediate preceeding nodes.

deowl · Answer

Yep, the OA is 20. 
Well done!

However I posted it to remind the (in)famous question from OG that deals
with the number of paths from A to B. So the quickest way to find the number of paths is to find the number of permutations of RRRLLL ( R & L for left and right ). 6!/(3!3!) = 20

giddi77 · Answer

Yep you are absolutely right. That's the quickest way of solving this question. :good

Zooroopa · Answer

giddi - your explanation is very good. However would this method help us to solve it within 2 minutes if the diamond shaped structure has more than 7 elements? Do you have a shorted method?

deowl - could you please elaborate a little on your explanation? I understand that we have six elements here once you freeze a vertex. How are you arriving at 6!/3!3!?

deowl · Answer

Please take a look at Q316 from OG10. This question is essentially the same. Obviously any path from the top to be bottom is composed of three 
steps to the left and three steps to the right. That is, let the word RRRLLL denote a unique path from the top to the bottom. Any permutation of the letters will also denote a unique path ( consider LLRRLR or LRLRLR ).
So all you need to find is the number of such permutations: 6!*(3!*3!) = 20

Futuristic · Answer

I'd go with deowl's explanation from the test point of view. Note that in these types of problems, it is usually the case that you have to find the shortest path from start to end. If this is not the case, one cannot solve the problem by this method. In this problem, it is implicitly stated; you cannot form the word without 3 steps to the left and 3 steps to the right.

We can see that the paths must be distinct, so permutations are what we seek. Remember the formula, where we find the number of permutations of letters in a word, where some letters are repeated? For example, LEVEL can be arranged in 5!/2!2! ways. The 2! comes in the denominator because of L being present twice, again for E being present twice.

Consider the organization of RRRLLL the same way in this problem. The number of permutations is 6!/3!3!

Zooroopa · Answer

Very good explanations - deowl and Futuristic. I think the key here is to comprehend that RRRLLL is the order that we want. I understand the explanation now. This is superior to the other explanation provided.

I extrapolated this to a diamond-shaped figure with more elements in it. I think deowl's method would solve the problem for any number of elements in it.

Thanks doewl and Futuristic!

PS: Harmony

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