1. ## pascal terms

ok so i'm using the notation (n r) for a pascal term

state numbers a, b and c such that
a*(8 5)=b*(8 6)=c*(7 5)

i thought this should be easy: i made three separate equation and developed the pascal terms into factorials like (8 5) = 8!/5!*3! = 8x7x6/2x3 but i just end up with stuff i don't know how to develop. how would you do this? i think there's just something simple im missing

2. I think that, by "Pascal term", you mean what I would call a "binomial coefficient; the numbers in Pascal's triangle, $\displaystyle \begin{pmatrix}n \\ m \end{pmatrix}= \frac{n!}{m!(n- m)!}$

Yes, $\displaystyle \begin{pmatrix}8 \\ 5\end{pmatrix}= \frac{8(7)(6)}{3(2)}= 42$
$\displaystyle \begin{pmatrix}8 \\ 6\end{pmatrix}= \frac{8(7)}{2}= 14$ and
$\displaystyle \begin{pmatrix}7 \\ 5\end{pmatrix}= \frac{7(6)}{1}= 42$

so your equation is 42a= 14b= 42c. That is two independent equations (42a= 14b and 42a= 42c; they imply 42a= 42c but this is not independent.) Those reduce to 3a= b and a= c. Unless you have additional information you have not told us there are an infinite collection of solutions. Choose any number for a. Then b= 3a and c= a. For example, if you arbitrarily choose a= 3, b= 9 and c= 3 satisfy the equations. If you choose a= 4.3, b= 12.9 and c= 4.3 also satisfy the equations.

3. i think you got (7 5) wrong, its 42
this is all the info i have
i have the answers and they are a, b, c are all different, i.e. not equal to one another, and they are specific, no infinite collection
but i'd like to know how to get them, i don't care for plain answers