Question 57

When two people cross the bridge at once, they can only travel as fast as the slowest person.

The crossings which should take place are:

• Anne and Bill cross the bridge $\left(2\text{ minutes} \right)$
• Anne returns with the torch $\left(1\text{ minute} \right)$
• Chen and Dwayne cross the bridge $\left(10\text{ minutes} \right)$
• Bill returns with the torch $\left(2\text{ minutes} \right)$
• Anne and Bill cross the bridge $\left(2\text{ minutes} \right)$

If the farmer closes $0$ gates, there is only $1$ enclosed area in which to place the cow.

If the farmer closes $1$ gate, there are ${6}\choose{1}$ $= 6$  ways to choose the gate, and in each case there are $2$ possible enclosed areas in which to place the cow.

If the farmer closes $2$ gates, there are ${6}\choose{2}$ $= 15$  ways to choose the gates, and in each case there are $3$ possible enclosed areas in which to place the cow.

Continuing this process, the total number of ways