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**+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

CPhill
Nov 24, 2018

#2**+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**+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**+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

#6**+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**+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**-1 **

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

I thought you know what n choose k means

Guest Nov 24, 2018