Louis has m shirts , n trousers and r caps , show that Louis has m × n × r different outfit combination without using a tree diagram.
How?
draw a tree
there are m branches on the first level
for each of those nodes, draw n branches for trousers.
On the 3rd level, draw r branches for each trousers node
I mean, how else can a tree look?
They said without a tree diagram
To show that Louis has m × n × r different outfit combinations, we can use the principle of multiplication.
First, let's consider the number of options for shirts. Louis has m shirts, so he can choose any one of them as his first choice. For each choice of shirt, he still has n options for choosing trousers. This means that for each of the m shirt choices, he has n options for trousers.
Now, for each combination of shirt and trouser, Louis still has r options for choosing a cap. So, for each combination of shirt and trouser, Louis has r options for the cap.
By the principle of multiplication, the total number of outfit combinations is the product of the number of choices at each step. Therefore, the total number of outfit combinations is m × n × r, as required.
In conclusion, Louis has m × n × r different outfit combinations without using a tree diagram.