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In this square array of 16 dots, four dots are to be chosen at random. What is the probability that the four dots will be collinear? Express your answer as a common fraction.
 

(someone put in a picture of a 4 by 4 square of dots here)

tanmai79  Nov 24, 2018
 #1
avatar+92856 
+3

Here's my best attempt, tanmai....

 

o       o         o         o

 

o      o         o          o

 

o      o         o          o 

 

o      o         o          o   

 

We have   C(16, 4)   = 1820   possible sets of any four of the points

 

But.....the only ones that will be collinear points  are the  four horizontal and vertical patterns and the two diagonal patterns   =   10 total

 

So....the probability   is      10 / 1820   =   1 / 182

 

 

cool cool cool

CPhill  Nov 24, 2018
 #2
avatar+135 
+2

You got it right!

 

I've got another question, though:

 

\(${16 \choose 4}$\)

 

What does that mean? There is no line to symbolize that it is a fraction.

tanmai79  Nov 24, 2018
 #3
avatar+3478 
+2

It's not a fraction. However, it means 16 choose 4; it computes the number of ways to choose items from a collection of items. 

tertre  Nov 24, 2018
 #4
avatar+3478 
+2

Also, a quote from CPhill's explanation: "We have   C(16, 4)   = 1820   possible sets of any four of the points."

 

We can express this as \(C\binom{16}{3}\) , where it is combinations. 

tertre  Nov 24, 2018
 #5
avatar+135 
+1

So is there, like, a formula or something to compute that?

tanmai79  Nov 24, 2018
 #6
avatar+3478 
+1

Oh yes, there is! Here: \(\frac{n!}{r!(n-r)!}\)- combinations; order does not matter

 

The order does matter( Permutations): \(\frac{n!}{(n-r)!}\)

 

If you take AoPS, they give you wonderful examples! 

tertre  Nov 24, 2018
 #10
avatar+135 
+2

I do take AoPS, but it's only the beginning of the year. Counting and Probability is the last 10 weeks of class.

tanmai79  Nov 24, 2018
 #11
avatar
-1

https://web2.0calc.com/questions/strange-q#r5

 

 

I thought you know what n choose k means

Guest Nov 24, 2018
 #12
avatar
-1

wait, why is this downvoted? I thought he knew what n choose k means because he used it before sad

Guest Nov 24, 2018
 #13
avatar+135 
+1

I just don't understand it well.

tanmai79  Nov 24, 2018
 #14
avatar
-1

why are you downvoting me tho

Guest Nov 24, 2018
 #15
avatar+135 
+1

It's not me downvoting you. Which Guest are you, Guest #1, Guest #2, or a guest I haven't met yet?

tanmai79  Nov 24, 2018
 #7
avatar+92856 
+2

Thanks, tertre     [ and tanmai ]   !!!  

 

 

cool cool cool          

CPhill  Nov 24, 2018
edited by CPhill  Nov 24, 2018
 #8
avatar+3478 
+2

Thank you, CPhill, for providing a wonderful solution! I was actually stumped at first..

tertre  Nov 24, 2018
 #9
avatar+92856 
+1

I just got lucky.....LOL!!!!

 

cool cool cool

CPhill  Nov 24, 2018

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