Question

A party of $9$ persons are to travel in two vehicles, one of which will not hold more than seven and the other not more than four. In how many ways can the party travel?

Answer 1

Here we need to take pairing, you can put person in the bus in following ways
$5$ in bus A and $4$ in bus B ${=}^9{C}_4$ ways
$6$ in bus A and $3$ in bus B ${=}^9{C}_3$ ways
$7$ in bus A and $2$ in bus B ${=}^9{C}_2$ ways
So, total number of ways would be ${(}^9{C}_4{+}^9{C}_3{+}^9{C}_2).$