($k_1,k_2$)-runs have received a special attention in the literature and its distribution can be obtained using combinatorial method (Huang and Tsai) and Markov chain approach (Dafnis et al). But the formulae are difficult to use when the number of Bernoulli trials is too large under identical setup and is generally intractable under non-identical setup. So, it is useful to approximate it with a suitable random variable. In this paper, it is demonstrated that pseudo-binomial is most suitable distribution for approximation and the approximation results are derived using Stein's method. Also, application of these results is demonstrated through real-life problems. It is shown that the bounds obtained are either comparable to or improvement over bounds available in the literature.