My task is to first show that the binomial pdf belongs to the exponential family, and then to identify the minimal sufficient statistic when $0<p<1$ is unknown (nothing is said about $n$ so I'm assuming that it is known).
First, I rewrite the pdf $f(x|\theta)={n \choose x} p^x(1-p)^{n-x}$ as $f(x|\theta)={n \choose x}e^{xln(\theta)+(n-x)ln(1-\theta)}$. From here, I can use the definition of a k-parameter exponential family and assert that if $a(\theta)=1$, $g(x)={n \choose x}$, $R_1(x)=x$, $b_1(\theta)=ln(\theta)$, $R_2(x)=(n-x)$, and $b_2(\theta)=ln(1-\theta)$, then $f(x|\theta)$ belongs to the 2 parameter exponential family.
Accordingly, the statistic $T=(\sum_{i=1}^nx, \sum_{i=1}^n(n-x))$ would be minimally sufficient. However, $T_2$ is just a 1-1 function of $T_1$, so this can't be minimally sufficient. Does anyone see where I may have messed up?