A corporation has seven available positions YI, Y2, ... , Y7 and there are ten applicants Xl, X2, . .. , XIO. The set of positions each applicant is qualified for is given, respectively, by {YI, Y2, Y6}, {Y2, Y6, Y7}, {Y3, Y4}, {Yl, Y5}, {Y6, Y7}, {Y3}, {Y2, Y3}, {YI, Y3}, {yd, {Y5}. Determine the largest number of positions that can be filled by the qualified applicants and justify your answer