I am confuse how to solve this.
Consider a bar consisting of n numbered squares
(see the figure on the right). You are to break the bar into smaller ones,
each of which must contain one or more complete numbered squares.
(1) How many different bars
can be obtained, including the original bar (10 points)?
(2) How many possible ways are there
for doing the division (5 points)?
Extending the bar to be an n × m bar formed by nm uniquely
numbered squares. We are to obtain smaller rectangular bars consisting of adjacent squares.
(3)
How many different bars can be obtained, including the original bar (5 points)? [Hint: for the first
question, think about one different bar at a time and how a unique bar may be obtained.]
Maybe I just dont understand the question but please help
Hint
(1) We have the one full bar of length $n$. Then there's the two bars of length $n-1$, then there's the three bars of length $n-2 \dots$ and finally we have the $n$ bars of length $1$.
(2) We can choose to cut the bar up to $n$ individual pieces, so that means we have a total of $n-1$ allowed places we can cut. Think of how many ways we can choose to cut $0\le i \le n-1$ slots out of the total $n-1$ slots.
(3) This one it would help to draw a picture. Notice thay if you have a big $n\times m$ bar, we can make different sized bars depending on the boundary lines we pick. Think about how many vertical dividing lines and horizonal dividing lines there are (including the outside edges), and to create a bar, we need to pick two of each...
See if this gets you in the right mindset in order to continue on your own