Question

Pairs of random numbers (x,y) are generated where the variables X and Y are integers between 0 to 5 inclusive. (i) How many different pairs are possible?

Answer 1

The answer is 15.

Explanation:

There are 6 possible integers from 0 to 5, so the n becomes 6. from these two we are supposed to make pair of 2 numbers do r becomes 2. By theory of permutation and combination, total number of ways in which we can draw combination of 2 integers from set of 6 integers is `=^6C_2` given by-

`=^6C_2`= `(6!)/(2!(6-2)!`= 15

Answer 2

i. `color(blue)(36)` pairs are possible
ii. distribution of `w=abs(x-y)`
`color(white)("XXX"){:(w=,0,1,2,3,4,5),("occurences:",6,10,8,6,4,2):}`

Explanation:

Part i
There are `color(green)6` possible values for `x`
and for each of these values there are `color(magenta)6` possible values for `y`

To be explicit:
#{: (,," | ",y=,,,,,), (,,ul(color(white)("xxx")),ul(0),ul(1),ul(2),ul(3),ul(4),ul(5)), (x=,0," | ",(0,0),(0,1),(0,2),(0,3),(0,4),(0,5)), (,1," | ",(1,0),(1,1),(1,2),(1,3),(1,4),(1,5)), (,2," | ",(2,0),(2,1),(2,2),(2,3),(2,4),(2,5)), (,3," | ",(3,0),(3,1),(3,2),(3,3),(3,4),(3,5)), (,4," | ",(4,0),(4,1),(4,2),(4,3),(4,4),(4,5)), (,5," | ",(5,0),(5,1),(4,2),(5,3),(5,4),(5,5)) :}#

Part ii
An examination of the table above should make the distribution of `w=abs(x-y)` obvious
or you could redo the table to be explicit:
#{: (w=abs(x-y),," | ",y,,,,,), (ul(color(white)(w=abs(x-y))),,ul(color(white)("xxx")),ul(0),ul(1),ul(2),ul(3),ul(4),ul(5)), (x=,0," | ",0,1,2,3,4,5), (,1," | ",1,0,1,2,3,4), (,2," | ",2,1,0,1,2,3), (,3," | ",3,2,1,0,1,2), (,4," | ",4,3,2,1,0,1), (,5," | ",5,4,3,2,1,0) :}#