Feature interaction modeling is crucial for deep recommendation models. A common and effective approach is to construct explicit feature combinations to enhance model performance. However, in practice, only a small fraction of these combinations are truly informative. Thus it is essential to select useful feature combinations to reduce noise and manage memory consumption. While feature selection methods have been extensively studied, they are typically limited to selecting individual features. Extending these methods for high-order feature combination selection presents a significant challenge due to the exponential growth in time complexity when evaluating feature combinations one by one. In this paper, we propose $\textbf{TayFCS}$, a lightweight feature combination selection method that significantly improves model performance. Specifically, we propose the Taylor Expansion Scorer (TayScorer) module for field-wise Taylor expansion on the base model. Instead of evaluating all potential feature combinations' importance by repeatedly running experiments with feature adding and removal, this scorer only needs to approximate the importance based on their sub-components' gradients. This can be simply computed with one backward pass based on a trained recommendation model. To further reduce information redundancy among feature combinations and their sub-components, we introduce Logistic Regression Elimination (LRE), which estimates the corresponding information gain based on the model prediction performance. Experimental results on three benchmark datasets validate both the effectiveness and efficiency of our approach. Furthermore, online A/B test results demonstrate its practical applicability and commercial value.