Re: Arrow AB which is a line segment exactly 5 units along with
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04 Mar 2021, 07:24
Hello!
We can solve this question by using Combinatorics as well.
My approach for this question is below:
Given two points with their co-ordinates: A(x1, y1), B(x2, y2). Then the length of AB should be:
AB^2 = 5^2 =25 = (x1-x2)^2 + (y1-y2)^2 = X^2 + Y^2 in which I denote X = x1-x2 and Y = y1-y2
Our goal is to determine X and Y such that X^2 + Y^2 = 25. Let's find out all the pairs (X^2, Y^2) that meet such a requirement, and also keep in mind that the square root of Y^2 and X^2 has to be integers. One example below does not meet this requirement such as the pair (1,24) as square root of 24 is not an integer:
(X^2,Y^2) = { (0,25), (1,24), ..., (9,16),...} Feel Free to exploit the whole set but we have found only one candidate in this list that meets the requirements above: (9,16) = (3^2, 4^2)
Next, we need to find out all the pairs (x1, x2) that has a difference of 3 and all the pairs (y1,y2) that has a difference of 4.
1. All the pairs (x1, x2) that has a difference of 3: (0,3), (1,4), ..., (6,9) and you flip the coordinates = 14 pairs
AND all the pairs (y1,y2) that has a difference of 4: (0,4), (1,5),...,(5,9) and you flip the coordinates = 12 pairs
2. Question: How many ways for you to choose a pair from 7 pairs AND a pair from 6 pairs? 14 x 12 = 168
Next: Don't forget, there are 2 ways to arrange 2 chose pairs once we choose one from (y1,y2) and one from (x1,x2) so we have a total of 168 x 2 = 336
3. Do the same thing for special case: AB^2 = 5^2 =25 = (x1-x2)^2 + (y1-y2)^2 = X^2 + Y^2 = 0 + 25
Find All the pairs (x1, x2) that has a difference of 0: (0,0), (1,1), ..., (9,9) and you do not flip the coordinates in this case as you will get identical pairs = 10 pairs
AND all the pairs (y1,y2) that has a difference of 5: (0,5), (1,6),...,(4,9) and you flip the coordinates = 10 pairs
4. Question: How many ways for you to choose a pair from 10 pairs AND a pair from 10 pairs? 10 x 10 = 100
Next: Don't forget, there are 2 ways to arrange 2 chose pairs once we choose one from (y1,y2) and one from (x1,x2) so we have a total of 100 x 2 = 200
The FINAL Answer is: 336 + 200 = 536