Please how is the combination addition formula ${{t}\choose{r}}={{t-1}\choose{r}}+{{t-1}\choose{r-1}}$ useful in proving the difference equation $\Delta_{t}{{r+t}\choose{t}}={{r+t}\choose{t+1}}$?
Secondly, does ${{t}\choose{k}}={{t}\choose{t-k}}$ need verification? I thought definition only testifies this.
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Just write your difference equation explicitely: $$\Delta_t \binom{r+t}{t}=\binom{r+t+1}{t+1}-\binom{r+t}{t} = \binom{r+t}{t+1}.$$
The identity $\binom{x}{y}=\binom{x}{x-y}$ is obvious if you defined binomial coefficients via factorials. If you have a combinatoric definition, then it's an evident exercise.
The difference equation is an immediate consequence of the addition formula:
$$\Delta_t\binom{r+t}t=\binom{r+t+1}t-\binom{r+t}t=\binom{r+t}{t-1}\;,$$
since
$$\binom{r+t+1}t=\binom{r+t}t+\binom{r+t}{t-1}\;.$$
Whether the identity $\binom{t}k=\binom{t}{t-k}$ needs proof depends on how you defined the binomial coefficient. If you defined it in terms of factorials, virtually no proof is required. If you defined it combinatorially, or as
$$\binom{t}k=\frac{t^{\underline k}}{k!}\;,$$
then some argument is required.