Answer:
see the attachment for the Venn diagram
- 1794
- 917
- 318
- 2153
- 248
Step-by-step explanation:
Given the numbers of students in various combinations of three courses, you want the associated Venn diagram and some numbers associated with additional combinations of courses.
Venn diagram
It is often easiest to populate the numbers in a Venn diagram by starting with the center one: 286 are taking all three courses. (V. = 286)
Working outwards from there, the numbers taking two courses can be filled in: (II. = History & Dance - V. = 343; IV. = History & Math - V. = 319; VI. = Dance & Math - V. = 318.)
Finally, the number taking a single course can be found. (I. = History -II. -IV. -V.; III. = Dance -II. -V. -VI.; VII. = Math -IV. -V. -VI.)
Adding all the numbers found so far gives the total in History, Dance, or Math (question 4). The difference between the number surveyed and the number in courses is VIII, the number taking none of the courses (question 5).
History or Not Dance
The least arithmetic may be involved if you rewrite this expression to ...
H ∪ D' = (H'∩D)' = 2401 -(III +VI) = 2401 -(289 +318) = 1794
(History or Math) and Not Dance
This will be the sum of numbers I, IV, VII.
226 +319 +372 = 917
Dance and Math and Not History
This is VI alone: 318
History, Dance, or Math
This is all but VIIII: 2401 -248 = 2153
None of the Courses
This is VIII alone: 248
