To graduate, John needs to complete eight courses. Of these eight, he must take only three science courses, only two...

He needs to take 8 courses in total, with 3+2+1=6 from the stated curriculum, leaving 2 other courses to be taken outside the stated curriculum.

There are 5+6+4=15 courses available from the stated curriculum and 20 courses in total. So the 2 courses to be taken outside the curriculum can be selected in 5!/2!3!= 10 ways.

The 3 science courses can be selected from the 5 available 5!/2!3!= 10 ways

The 2 math courses selected from 6 available 6!/2!4!=15 ways

The 1 history course from the 4 available 4!/3!1!= 4 ways

Total ways = 10*10*15*4=6000,D