\ (n = 0, n = 1 \) when apparently established
\ (n \ geq2 \) , it is assumed \ (n-1, n- 2 \) are true, then:
\ [\ sum_ {I = K} ^ n-\ tbinom {I} {K} = \ tbinom {n- } {k} + \ sum_ { i = k} ^ {n-1} \ tbinom {i} {k} = \ tbinom {n} {k} + \ sum_ {i = k} ^ {n-1} \ tbinom {i} {k} = \ tbinom {n} {k} + \ tbinom {n} {k + 1} = \ tbinom {n + 1} {k + 1} \]