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# Fascinating Lattices Problem

Find the number of ways in which 20 $$\alpha$$ and 20 $$\beta$$ can be arranged in a row such that upto any point in the row, number of $$\alpha$$ is more than or equal to number of $$\beta$$.

Note: You may use calculator, if required.

I posted this question one or two weeks ago (as a problem)! But got no response! So I thought of deleting it and reposting it here as a note. So that some real 'geniuses' can help me solve it out.

Note by Pranjal Jain
2 years, 2 months ago

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This is a fascinating lattices problem. Here is a diagram that might help!

Imgur

· 2 years, 2 months ago

@Pranjal Jain If you don't understand what I'm talking about, think of ordering the 40 characters like moving on a grid. A step north is like adding an $$\alpha$$ to the sequence, and a step east is like adding a $$\beta$$. · 2 years, 2 months ago

1) Pic is not appearing 2) I tried modelling it that way but didnt reached answer. · 2 years, 2 months ago

Also, I calculate the answer to be 6564120420. · 2 years, 2 months ago

Correct!! It comes out to be $$\frac{\dbinom{2n}{n}}{n+1}$$ for n $$\alpha$$s and n $$\beta$$s. · 2 years, 2 months ago

Exactly! It's a Catalan number. :D · 2 years, 2 months ago

Eh!? You seem much more awesome and interesting than what I imagined! · 2 years, 2 months ago

Haha, thank you. Have you ever heard of Bertrand's ballot theorem? You NEED to check it out if you like this type of problem. It's one of the most beautiful theorems in my opinion. :D · 2 years, 2 months ago

Couldn't sleep!! I read complete wikipedia page of that Ballot theorem! You were correct... One of the most beautiful theorems! Any wiki page for it here on brilliant? · 2 years, 2 months ago

@Pranjal Jain No, a Wiki page does not exist for the Ballot Theorem as yet. Can you add one?

@Finn Hulse Note that to display an image, you have to link to the image file (typically ending with .png, .jpg .gif etc), as opposed to linking to an entire site. Staff · 2 years, 2 months ago

I have a camp at Kolkata. So i'll be busy next week. I will add one as soon as I reach home back! · 2 years, 2 months ago

Oh! Ill check it out after a sleep! Its 5:30AM in India. Lol! Going to sleep! · 2 years, 2 months ago

Yeah... That's strange because I can't see it either. The link is http://imgur.com/5Nxc7RN. · 2 years, 2 months ago