Math

There are 15 applicants for a job. In how many ways can 10 jobs be filled if only 5 of the applicants are qualified receptionists, 7 of the
applicants are qualified filing clerks, and only 3 of the candidates are suitable for both a receptionist position and a filing clerk position? All
applicants can fill any other position that is available. Every position that is available is unique. You must hire one applicant for the filing
clerk position and one applicant for the receptionist position.

The answer's supposed to be 1 660 538 880.

I already tried doing it but I keep getting it wrong. Since it says all positions are unique I assumed that it uses permutations and used cases to try and solve it:

Case 1: The applicants who have both qualifications take both positions
3P2 x 13P8
Case 2: Applicant with both qualifications takes only clerk position
3P1 x 5P1 x 13P8
Case 3: Applicant with both qualifications takes only reception position
3P1 x 7P1 x 13P8
Case 4: Single qualification applicants take the positions
7P1 x 5P1 x 13P8

The answer I get is much larger. I don't know what I'm doing wrong. . .

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  1. Oh wait. I realized what I did wrong. I forgot that there's overlap between the applicants and that the double qualification applicants also get counted into the other two.

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