Problem Analysis:
We need to arrange nine boys and five girls in two vans such that at least four girls always sit together. The vans have three seats in the front and five seats at the back.

Approach:
To solve this problem, we can use the concept of permutations and combinations. Let's break down the problem into smaller steps:

1. Calculate the total number of ways to arrange the nine boys and five girls in two vans without any restrictions.
2. Calculate the total number of ways to arrange the nine boys and five girls in two vans with all the girls sitting together.
3. Subtract the number of ways calculated in step 2 from the number of ways calculated in step 1 to get the final answer.

Step 1: Total Number of Ways to Arrange Boys and Girls:
We have nine boys and five girls, which means we need to arrange a total of 14 people. In each van, there are eight seats in total (three in the front and five at the back). Therefore, the total number of ways to arrange the boys and girls in the two vans without any restrictions can be calculated using the formula for permutations:

Total number of ways = 14P8 = 14! / (14-8)!

Step 2: Number of Ways to Arrange Girls Together:
To calculate the number of ways to arrange the girls together, we can consider them as a single entity. Therefore, we have five entities (four girls together and one group of boys) to arrange in two vans. Again, using the formula for permutations, we can calculate the number of ways as:

Number of ways = 5P2 = 5! / (5-2)!

Step 3: Final Answer:
Finally, we subtract the number of ways calculated in step 2 from the number of ways calculated in step 1 to get the final answer:

Final Answer = Total number of ways - Number of ways to arrange girls together

Calculation:
Let's calculate the final answer using the formulas for permutations:

Total number of ways = 14P8 = 14! / (14-8)! = 14! / 6! = 3003

Number of ways to arrange girls together = 5P2 = 5! / (5-2)! = 5! / 3! = 20

Final Answer = Total number of ways - Number of ways to arrange girls together = 3003 - 20 = 2983

Therefore, there are 2983 ways in which nine boys and five girls can be arranged in two vans such that at least four girls always sit together.

Conclusion:
In this problem, we used the concepts of permutations and combinations to calculate the number of ways in which nine boys and five girls can be arranged in two vans such that at least four girls always sit together. By considering the arrangements of the girls as a single entity, we were able to calculate the desired answer. The final answer is 2983 ways.