Query about how the domain of a binomial coefficient was calculated

[Snowman2]

Homework Statement

Not a homework question per-se, but this is the question which lead to the confusion

Find the domain of the function:
i) C(16-x , 2x-1)
(Sorry I did not know how to write the superscript before C aha!, just saw a standard notation and apparently we can write it like how I wrote above!)

Relevant Equations

So it has been a while since I have been in school, but I just picked up one of these elementary calculus books to brush up my basics and I came across this question:

The solution the author provided for C(n,r) to be defined was
i)n>0
ii) r should be 0<=r<=n
iii) n & r should be integers.

He writes x<16 & x>=0.5 & x<=17/3
I agree
Then he writes x€ [0.5,17/3]
I agree though not the whole interval obviously
Then he directly writes x={1,2,3,4,5}
I am unable to understand why he writes only integer values for x, the definition said the superscript and subscript should be integers, not x ?

Also if I need to find all the values of x where the n and r in my question becomes an integer how am I supposed to do that? Do I input all of these values that lie in x's interval into the n and r expression to see if it is an integer? Wouldn't it be very ugly and not so smart?

Smart people please let me know

PS: This is my first post so I am not very sure with the guidelines and where to post what. I saw math I clicked math. Please let me know if I did something wrong.

 

[SammyS]

Homework Statement: Not a homework question per-se, but this is the question which lead to the confusion

Find the domain of the function:
i) C(16-x , 2x-1)
(Sorry I did not know how to write the superscript before C aha!, just saw a standard notation and apparently we can write it like how I wrote above!)
Relevant Equations:

So it has been a while since I have been in school, but I just picked up one of these elementary calculus books to brush up my basics and I came across this question:

The solution the author provided for C(n,r) to be defined was
i)n>0
ii) r should be 0<=r<=n
iii) n & r should be integers.

He writes x<16 & x>=0.5 & x<=17/3
I agree
Then he writes x€ [0.5,17/3]
I agree though not the whole interval obviously
Then he directly writes x={1,2,3,4,5}
I am unable to understand why he writes only integer values for x, the definition said the superscript and subscript should be integers, not x ?

Also if I need to find all the values of x where the n and r in my question becomes an integer how am I supposed to do that? Do I input all of these values that lie in x's interval into the n and r expression to see if it is an integer? Wouldn't it be very ugly and not so smart?

Smart people please let me know

PS: This is my first post so I am not very sure with the guidelines and where to post what. I saw math I clicked math. Please let me know if I did something wrong.

Hello @Snowman2 .


Isn't it true that $16-x$ must be an integer ?

 

[Snowman2]

Hello @Snowman2 .


Isn't it true that $16-x$ must be an integer ?

Hello @SammyS

Yes but for 2x-1 how can I be sure that the integer values of x are going to be the only values which give me 2x-1 as an integer?

For eg I could have something like x=1/2 which gives 2x-1 as an integer and x here is a real number? I understand that if we intersect this with the set of values of x which give us 16-x as an integer, we would not get 1/2, infact we would only get integers as stated.

But the author goes on to generalise this by saying this is a valid approach which would work for all sums concerning a binomial coefficient's domain. Perhaps I should have mentioned that in the post

Anyways let the sun rise i will also post a snapshot of that claim from my book.
Till then
Cheers!