Of a company's personnel, seven work in design, 14 in manufacturing, four in testing, five in sales, two in accounting, and three in marketing. A committee of six people is to be formed to meet with upper management. In how many ways can the committee be formed if there must be at least two members from the manufacturing department?
Am I double counting if I have $\left.14\choose 2\right.\left.33\choose 4\right.$? The logic I'm using is that once your 2 people are chosen for the two required manufacturing seats, we don't care what other departments fill the other 4 seats. But I've been told in some other problems (of which this seems similar) that I could be double counting if I do it like this.