Many previous research studies on extractive text summarization consider a subset of words in a document as keywords and use a sentence ranking function that ranks sentences based on their similarities with the list of extracted keywords. But the use of key concepts in automatic text summarization task has received less attention in literature on summarization. The proposed work uses key concepts identified from a document for creating a summary of the document. We view single-word or multi-word keyphrases of a document as the important concepts that a document elaborates on. Our work is based on the hypothesis that an extract is an elaboration of the important concepts to some permissible extent and it is controlled by the given summary length restriction. In other words, our method of text summarization chooses a subset of sentences from a document that maximizes the important concepts in the final summary. To allow diverse information in the summary, for each important concept, we select one sentence that is the best possible elaboration of the concept. Accordingly, the most important concept will contribute first to the summary, then to the second best concept, and so on. To prove the effectiveness of our proposed summarization method, we have compared it to some state-of-the art summarization systems and the results show that the proposed method outperforms the existing systems to which it is compared.