Spm test question(Combination)

Question

$\binom{y}{m}=\binom{y}{n}$,

How should I express y in terms of m and n?

Answer 1

There is an identity that

If $n\choose m$$=$$n\choose a$ then either $a=m$ or $a+m=n$ . In your case $y=n+m$ .

Answer 2

It is important to note that

$${y \choose m} = {y \choose n} \iff \color{blue}{m+n = y}$$

Unless $m = n$, which is trivial. You cannot express $y$ in terms of $m$ and $n$ in that case because $y$ could be any positive integer.

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This can be shown very easily.

$${y \choose m} = {y \choose n}$$

Substituting $\color{blue}{m = y-n}$, you get

$${y \choose \color{blue}{y-n}} = {y \choose n}$$

$$\frac{y!}{(y-n)!(y-(y-n))!} = \frac{y!}{n!(y-n)!}$$

$$\frac{y!}{(y-n)!n!} = \frac{y!}{n!(y-n)!}$$

Which is true. You can also prove this backwards.

$$\frac{y!}{m!(y-m)!} = \frac{y!}{(y-m)!(y-(y-m))!}$$

$${y \choose m} = {y \choose y-m}$$