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In how many ways we can select two numbers from first $10$ natural numbers so that difference between them is at least four?

If we select the numbers from the interval $[1,10]$, will the answer change? Here $[1,10]$ means the interval containing all real numbers from $1$ to $10$. I have tried it using geometrical probability.

N. F. Taussig
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user344005
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  • Is 6 - 10 valid ? –  Jun 02 '16 at 07:26
  • What do you mean by length[0,10]? – Hailey Jun 02 '16 at 07:26
  • $10$ ways to choose the first number, $6$ ways to choose the second number. – barak manos Jun 02 '16 at 07:27
  • @barakmanos only if we consider absolute value. –  Jun 02 '16 at 07:28
  • @barakmanos there should be less than $6$ ways to choose the second one, like if I start with $5$ – mastrok Jun 02 '16 at 07:29
  • (5,9) and (9,5) selections are same.right? – Hailey Jun 02 '16 at 07:30
  • You should provide more explanations (what's the length [1,10]? Is it "the natural numbers from 1 to 10" as opposed to "the natural numbers from 0 to 9"?) and also how you tried to solve this. Otherwise, it just smells of "I have this exercise for school, anyone wants to do it for me?", sorry. – polettix Jun 02 '16 at 07:46
  • @polettix how can 0 be a natural number? – Hailey Jun 02 '16 at 08:11
  • Thanks everyone..here [0,10] means interval containing all real number from 1 to 10.i have tried it using geometrical probability – user344005 Jun 02 '16 at 09:00
  • There are uncountably many real numbers in the open interval between the two chosen numbers, so the answer definitely changes. Please check that I did not change the meaning of your post when I edited it. If you want to calculate the geometric probability that two real numbers selected from an interval of length 10 that includes the first ten natural numbers are at least four units apart, you may want $[0.5, 10.5]$ or $(0, 10]$. I am leaving the left end of the interval $(0, 10]$ open since $0$ is considered to be a natural number by some mathematicians. – N. F. Taussig Jun 02 '16 at 09:50
  • @Hailey I remember my Geometry professor telling us that the inclusion of 0 is somehow disputed. See https://en.wikipedia.org/wiki/Natural_number and also https://en.wikipedia.org/wiki/0_%28number%29 that states that "By most definitions 0 is a natural number". Which leads us to the Peano axioms, whose first one is "0 is a natural number" (https://en.wikipedia.org/wiki/Peano_axioms), although "Today, some mathematicians prefer to replace "zero" by "one" in Peano's axioms" (https://books.google.it/books?id=7xArILpcndYC&pg=PA255&redir_esc=y#v=onepage&q&f=false). So... that's why I was asking! – polettix Jun 09 '16 at 12:41

1 Answers1

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There are ambiguities in your question, as many have pointed out.

I am working out a way for natural #s $1$ thru $10$, you can easily adapt it to whatever you had in mind.

Take $7$ unnumbered balls, and a block of $3$ balls.

Choose any two from the seven unnumbered balls in $\binom72 = 21$ ways,
insert the block of $3$ immediately after the leftmost of the two chosen balls,
now number the balls serially.

All such configurations will satisfy the stipulations, thus ans $=\fbox{21}$

true blue anil
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