  Binomial coefficients have been used for centuries in a variety of fields and
have accumulated numerous definitions. In this paper, we introduce a new way of
defining binomial coefficients as repeated sums of ones. A multitude of
binomial coefficient identities will be shown in order to prove this
definition. Using this new definition, we simplify some particular sums such as
the repeated Harmonic sum and the repeated Binomial-Harmonic sum. We derive
formulae for simplifying general repeated sums as well as a variant containing
binomial coefficients. Additionally, we study the $m$-th difference of a
sequence and show how sequences whose $m$-th difference is constant can be
related to binomial coefficients.
