A team of 8 students goes on an excursion, in two cars, of which one can seat 5 and the other only 4. In how many ways can they travel?

There are 8 students and the maximum capacity of the cars together is 9.

We may divide the 8 students as follows

Case I: 5 students in the first car and 3 in the second Or

Case II: 4 students in the first car and 4 in the second

Hence,     in Case I: 8 students are divided into groups of 5 and 3 in  $8C3$ ways.

Similarly, in Case II: 8 students are divided into two groups of 4 and 4 in  $8C4$ ways.

Therefore, the total number of ways in which 8 students can travel is

$8C3+8C4$ = 56 + 70 = 126.