If Miguel went to science club every Thursday for 4 months and three weeks and each Thursday 25 chairs were set up for club members to sit in, and about 15 club members and up came each Thursday, how many times would Miguel have sat in a seat that Clara had already sat in before?
If Miguel went to science club every Thursday for 4 months and three weeks, which is approximately 17 weeks, then he would have attended the club 17 times.
If each Thursday 25 chairs were set up for club members to sit in, then in total, there would have been 17 * 25 = <<17*25=425>>425 chairs available.
If about 15 club members and up came each Thursday, then the maximum number of club members Miguel could have sat with would have been 15.
If Miguel sat in a seat that Clara had already sat in once, then he would have sat in a seat that Clara had already sat in once for 15 times since he only sat with 15 club members max.
The difference in the number of seats Miguel sat in, that Clara had already sat in before, is 425 - (25 * 15) = 425 - 375 = <<425-(25*15)=50>>50 times. Answer: \boxed{50}.