10. Balls $A, B, C, D, E$ are to be put into seven boxes numbered 1 to 7 . Each box can contain at most one ball. Furthermore, it is required that the box numbers for balls $A$ and $B$ must be consecutive, and that the box numbers for balls $C$ and $D$ must also be consecutive. In how many different ways can the balls be put into the boxes?
(1 mark)
現要把球 $A 、 B 、 C 、 D 、 E$ 放進七個編號為 1 至 7 的盒子中, 其中每個盒子最多只可放一個球。此外, 球 $A$ 和 $B$ 必須被放進兩個編號連續的盒子，而球 $C$ 和 $D$ 亦必須被放進兩個編號連續的盒子。問有多少種方法把球放進盒子？
10. 240
10. We first choose 4 boxes to place balls $A, B, C, D$. There are 10 possible combinations, namely, $1234,1245,1256,1267,2345,2356,2367,3456,3467$ and 4567 . Once the 4 boxes are chosen, there are 4 ways to place ball $A$, and then 1 way to place ball $B$ (once ball $A$ is placed there is only one possible position for ball $B$ in order that balls $C$ and $D$ can be placed in boxes with consecutive numbers), and finally 2 ways to place balls $C$ and $D$. There are 3 remaining boxes in which ball $E$ can be placed. Hence the answer is $10 \times 4 \times 2 \times 3=240$.