Ruxandra wants to visit Singapore, Mongolia, Bhutan, Indonesia, and Japan. In how many ways can she order her trip to visit each country exactly once, with the conditions that she cannot visit Mongolia first and cannot visit Bhutan last?
Since Ruxandra wants to visit 5 countries, then there are $5!=5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=120$ orders in which she can visit them with no restrictions.

Of these ways, there are $1 \cdot 4 \cdot 3 \cdot 2 \cdot 1=24$ ways in which she visits Mongolia first. (Visiting Mongolia first is fixed and so there is 1 choice for the first country; there are still 4 choices for the second country, 3 for the second country, and so on.)

Similarly, there are 24 ways in which she visits Bhutan last.

Both of these totals of 24 include the ways in which she visits Mongolia first and visits Bhutan last. There are 6 such ways, since there is 1 choice for the first country and 1 choice for the
last country, and then 3 choices for the second country, 2 choices for the third country, and 1 for the fourth country.

Therefore, the number of ways in which she either visits Mongolia first or Bhutan last (or both) is $24+24-6=42$.

Therefore, the number of ways with the condition that she does not visit Mongolia first and she does not visit Bhutan last is $120-42=78$.

ANSWER: 78