Let 'n' stand for the number of multiple choice questions. We can write:

If Pam got 4 fill-in questions correct:

4(7)+n(3)=78

n=16.67 (not possible because not a whole number)

If Pam got 3 fill-in questions correct:

3(7)+n(3)=78

n=19 (possible answer because it is a whole number)

If Pam got 2 fill-in questions correct:

2(7)+n(3)=78

n=21.33 (not possible because not a whole number)

If Pam got 1 fill-in question correct:

1(7)+n(3)=78

n=23.67 (not possible because not a whole number)

Therefore, she must have gotten 19 correct because it is the only whole number answer.