How to prove if function is increasing Need to prove that function $$P(n,k)=\binom{n}{k}= \frac{n!}{k!(n-k)!}$$ is increasing when $\displaystyle k\leq\frac{n}{2}$.
Is this inductive maths topic?

As $n$ increases, $n!$ increases.  As $n$ increases $(n-k)!$ increases.  $(n+1)!$ is $(n+1)$ times larger than n!.  $(n+1-k)!$ is however only $(n+1-k)$ times greater than $(n-k)!$.  Therefore the numerator increases more than the denominator as n increases.  Therefore P increases as n increases.
As $k$ increases, $k!$ increases.  As $k$ increases $(n-k)!$ decreases.  The question is then, what changes at a greater rate: $k!$ or $(n-k)!$.  Define the functions $A(x) = x!$ and $B(x) = (n-x)!$.  The denominator of $P(n,k)$ is thus $A(k)*B(k)$.
Now $A(k) = k!$ and $B(k) = (n-k)!$.
Also, $A(k+1) = (k+1)! = (k+1)*k!$ 
Also, $B(k+1) = (n - k - 1)! = (n-k)!/(n-k)$. 
Therefore $A(k+1)B(k+1) = (k+1)k!(n-k)!/(n-k) = A(k)B(k)(k+1)/(n-k)$
So $(k+1)/(n-k) \leq 1$ then the function is increasing.
This must be the case since $k \leq n/2$  
Therefore P is an increasing function of $k$ and $n$.
