(1+x)^p=1+px+p(p-1)x^2/2!+...+pCrx^r+...+x^p.
If p=m+n, then the general term is (m+n)Crx^r and the last term is x^(m+n).
(1) a+b=m+n=p: the coefficient of the rth term (x^r) is p!/(r!(p-r)!)=pCr by definition. Example: if p=7 and r=3, this becomes 7!/(3!4!)=7*6*5/3*2*1. p!/(r!(p-r)!)=(m+n)!/(r!(m+n-r)!) is the coefficient of x^r. If r=m, then the coefficient of x^m is a=(m+n)!/(m!n!) and the coefficient of x^n is b=(m+n)!/(n!m!). Therefore, a=b (3). (Using the example, and putting m=3 and n=4 the coefficients are 1, 7, 21, 35, 35, 21, 7, 1. The coefficient of x^3 and x^4 are both 35. If we put m=5 and n=2 the coefficients of x^5 and x^2 are both 21. This follows the symmetry of Pascal's Triangle.)