Intresting combination & probability problem

I am a DOTA 2 player and I came across this intriguing problem which I do not know how to solve. In this game 1 match is played between two teams, each consisting of 5 players. Each player has to pick a hero/character from a pool of total 107 heroes, i.e 10 heroes picked. Moreover both team's captain bans 5 heroes for other team to pick. i.e 10 heroes banned.

Now, in a tournament of 16 teams, played on a knockout format with each contest between two teams consisting of 3 matches (best of 3 winner) and grand final of 5 matches. What is the probability that atleast X (lets say 5) heroes will not be picked or banned in whole tournament.

#Combinatorics #Probability

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

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

I don't know of any simple formula for this one, but here's what I came up with:

In the whole tournament, there will be $(8+4+2) \cdot 3 + 1 \cdot 5 = 47$ matches. That translates to $47 \cdot 20 = 940$ hero-selections (either for picking or banning, that isn't our concern).

If you don't want $y$ players to be selected in the whole tournament, then, effectively in every match you have $\displaystyle N_y = 107 - y$ heroes. So, in each match, you can have $\displaystyle {N_y \choose 20}$ selections. And since there are $940$ matches, totally there are $\displaystyle {N_y \choose 20}^{940}$ possible selections.

Thus, the probability of $y$ heroes being completely neglected in the whole tournament is $\displaystyle P(y) = \dfrac{ {N_y \choose 20}^{940} } { {107 \choose 20}^{940} }$ .

So, the probability that at least $X$ are totally neglected in the whole tournament is:

$\displaystyle 1 - \sum_{0 \le y \le {X-1} } P(y)$

Parth Thakkar - 7 years ago

And since there are $1140$ matches,

Isn't there only 14 matches. Other than that, good solution

Siddhartha Srivastava - 7 years ago

Thank you Parth, But i think ( 8 + 4 + 2 ) x 3 + 1 x 5 = 47...So, there will be 47 matches in total 15 contests..that would mean 47 x 20 = 940 hero-selections. Right?

A good solution though.

Musabbir Hussain - 7 years ago

Aah that was so dumb! :D Thanks for pointing it out. Dumb Dumb Dumb! Sometimes its fun to do dumb things though ;) :D Edited.

Parth Thakkar - 7 years ago

@Parth Thakkar – No Problem..I also do such dumb things alot :D..! another thing I would like to understand is that why did u put ( 940 20 ) in denominator..Can you please explain the equation in words. I am sorry if that is something too obvious. Sadly, I am much interested but not so good in Probability, Combination & Permutation :-S

Musabbir Hussain - 7 years ago

@Musabbir Hussain – That's another mistake I made! Well, it should be ${107 \choose 20}$ and not ${940 \choose 20}$ . That's the total number of possible selections, without any player being neglected.

Parth Thakkar - 7 years ago

One question. Can both teams ban the same hero?

Siddhartha Srivastava - 7 years ago

No. A hero can be picked/banned only once.

Musabbir Hussain - 7 years 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}`	$\boxed{123}$