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[Originally posted in Discussions]
I was wondering whether there was another way you could explain the multiplication principle so I can get a better intuitive understanding the principle and why if I have for example 3 choices for the first digit, and three for the second and three for the third, then the total number of ways that those numbers can be arranged is 27? Why are the different arrangements the result of multiplying the 3 choices by the 3 choices by the 3 choices? When I draw the tree diagram, I can see that is true but I am struggling to understand intuitively why the fact that I have 3 choices for the first number and three for the second etc is related to multiplying each number of choices by the next number of choices? Multiplication is simply repeated addition. Why is repeated addition the way we get the answer of the total number of rearrangements? Sorry if this question is not clear. I have a feeling my lack of feel for the multiplication principle is maybe my lack of understanding what multiplication actually means.
That's a really great question, and this principle is one of the reasons why counting and combinatorics can be so tricky. It's wonderful that you're trying to delve into the meaning of why we multiply the number of choices together. The basic idea is that multiplication represents groups, and the number of ways to combine different choices can also be represented using groups.
There is another question from Professor Loh's original Mock Mathcounts Sprint Round relating to this concept. There is a free video explanation of one of the questions, which happens to relate to the multiplication principle. In the video, there are three mini-questions, one of which asks to find the number of ways to choose 2 people out of 8 to stand side-by-side for a photo:
I'll copy the explanation again here:
The important idea here is to use multiplication, not addition, to find the ways to arrange the two people. Multiplication can be used to express the number of things that are arranged into groups.
Let's first draw 8 people and name them Person 1, Person 2, Person 3, Person 4, Person 5, Person 6, Person 7 and Person 8.
The ways to pick two of them to stand side-by-side for a picture can be arranged into eight groups, each of which has seven ways in it:
In the picture above, a ways described as "48" would mean that Person 4 stands on the left, and Person 8 stands on the right. Order matters here, because the photo would look different if Person 8 were on the left and Person 4 were on the right, which is described as "84."
There are eight groups because there are eight people who can stand on the left, and given that choice of person on the left, there are seven choices for the person on the right.
Please ask if you have any more questions!
The Daily Challenge Team