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.

#Combinatorics #Permutations #Combinations #Mathsfun

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Michael Mendrin @Calvin Lin @Finn Hulse @Trevor Arashiro @Trevor B. @megh choksi @Sandeep Bhardwaj @Satvik Golechha @Krishna Ar Please help

Pranjal Jain - 6 years, 7 months ago

@John Muradeli @Agnishom Chattopadhyay @Sharky Kesa @Daniel Liu @Christopher Boo @brian charlesworth you too help

Pranjal Jain - 6 years, 7 months ago

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

Imgur Imgur

Finn Hulse - 6 years, 7 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 $\beta$ .

Finn Hulse - 6 years, 7 months ago

1) Pic is not appearing 2) I tried modelling it that way but didnt reached answer.

Pranjal Jain - 6 years, 7 months ago

@Pranjal Jain – Yeah... That's strange because I can't see it either. The link is https://imgur.com/5Nxc7RN.

Finn Hulse - 6 years, 7 months ago

@Pranjal Jain – Also, I calculate the answer to be 6564120420.

Finn Hulse - 6 years, 7 months ago

@Finn Hulse – Correct!! It comes out to be $\frac{\dbinom{2n}{n}}{n+1}$ for n $\alpha$ s and n $\beta$ s.

Pranjal Jain - 6 years, 7 months ago

@Pranjal Jain – Exactly! It's a Catalan number. :D

Finn Hulse - 6 years, 7 months ago

@Finn Hulse – Eh!? You seem much more awesome and interesting than what I imagined!

Pranjal Jain - 6 years, 7 months ago

@Pranjal Jain – 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

Finn Hulse - 6 years, 7 months ago

@Finn Hulse – Oh! Ill check it out after a sleep! Its 5:30AM in India. Lol! Going to sleep!

Pranjal Jain - 6 years, 7 months ago

@Finn Hulse – 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?

Pranjal Jain - 6 years, 7 months ago

@Pranjal Jain – @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.

Calvin Lin Staff - 6 years, 7 months ago

@Calvin Lin – I have a camp at Kolkata. So i'll be busy next week. I will add one as soon as I reach home back!

Pranjal Jain - 6 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$