A ladder has 5 rungs. Elisabeth can climb up by 1 or 2 rungs at a time. In how many different ways can she climb up to the fifth rung of the ladder?
(A) 10
(B) 9
(C) 7
(D) 6
(E) 8
Elisabeth climbs a total of 5 rungs by climbing either 1 or 2 rungs at a time.

Since there are only 5 rungs, then she cannot climb 2 rungs at a time more than 2 times.

Therefore, she must climb 2 rungs either 0,1 or 2 times.

If she climbs 2 rungs 0 times, then each step consists of 1 rung and so she climbs 1,1,1,1,1 to get to the top.

If she climbs 2 rungs 1 time, then she climbs 2,1,1,1, since the remaining 3 rungs must be made up of steps of 1 rung.

But she can climb these numbers of rungs in several different orders.

Since she takes four steps, she can climb 2 rungs as any of her 1st, 2nd, 3rd, or 4th step.

Putting this another way, she can climb $2,1,1,1$ or $1,2,1,1$ or $1,1,2,1$ or $1,1,1,2$.

There are 4 possibilities in this case.

If she climbs 2 rungs 2 times, then she climbs 2,2,1.

Again, she can climb these numbers of rungs in several different orders.

Since she takes three steps, she can climb 1 rung as any of her 1st, 2nd or 3rd step.

Putting this another way, she can climb $1,2,2$ or $2,1,2$ or $2,2,1$.

There are 3 possibilities in this case.

In total, there are $1+4+3=8$ ways in which she can climb.

ANswer: (E)