Kieren MacMillan, Jonathan Sondow, Proofs of Power Sum and Binomial Coefficient Congruences Via Pascal’s Identity, The American Mathematical Monthly, Vol. 118, No. 6 (June‒July 2011), pp. 549-551 ...

We produce an alternate proof of the extended binomial theorem by solving a first order linear ordinary differential equation with a given initial condition. The method is easy to follow and the ...