The Multiplication Principle
Hi. Now that you’ve got some idea of counting related problems and use of multiplication to answers them, I’ll now state the Multiplication Principle a bit formally:
“If a collection of objects can be separated into m different types, and each of these types can be separated into n different subtypes, then there will be m x n different types of objects in all.”
(There is a reason why I’ve been bolding each each along the way, because each each matters and things will be different without them. I’ll come to this a little later.)
For our cars example, we could separate the cars into 2 types (sedan and hatchback) and each of these could be separated into 3 subtypes (models), making 2 x 3 = 6 total types of cars.
(We could also have gone the other way round and first separate the cars as models, and then further separate these into sedans and hatchbacks. The answer would have remained the same: 3 x 2 = 6)
Further, each of these 6 cars can be separated into 2 types (red and blue) making a total of 6 x 2 = 12 different types of cars.
The multiplication rule is not just limited to classification of objects. It can also be applied in different contexts. Consider this problem, for example – If there are 3 different flights from A to B and 2 different trains from B to C, in how many different ways can a person reach from A to C (using only these flights and trains)?
Here are all the cases for you to look at.
The possible routes are – 1a, 1b, 2a, 2b, 3a, and 3b – a total of 6.
Again, for each route from A to B, there are 2 routes from B to C. And since there are 3 routes from A to B, there will be a total of 3 x 2 different routes in all.
In other words, the number of ways will be obtained by multiplying the number of routes from A to B (3) with the number of routes from B to C (2) available for each route from A to B: 3 x 2 = 6.
Here’s another way we can state the multiplication principle:
“If a task T can be divided into subtasks T1 and T2, which can completed in m ways and n ways respectively, and T will be completed by completing both T1 and T2, then the number of ways of completing T will be m x n”
Let’s think of this example again.
The two subtasks T1 and T2 can be thought of as reaching from A to B and reaching from B to C respectively, where T1 can be completed in 3 ways and T2 can be completed in 2 ways.
Since the task of reaching from A to C will be completed by completing both T1 and T2, therefore the number of ways of reaching from A to C will be 3 x 2 = 6.
Here’s another simple problem.
In how many ways can you reach from A to D, given the possible routes from A to B, B to C and C to D?
The task here is A to D, which will be completed by completing the tasks A to B and B to D.
Here B to D can further be divided into subtasks B to C (2 ways) and C to D (3 ways), both of which must be completed. Therefore the number of ways of reaching from B to D is 2 x 3 = 6.
And the number of ways of completing A to D will be 3 (A to B) x 6 (B to D) = 18 ways.
We could also directly write this number as 3 (A to B) x 2 (B to C) x 3 (C to D) = 18 ways.
We can state the multiplication principle in the following way:
““If a task T can be divided into n subtasks T1,T2,T3,….Tn, which can completed in m1, m2, m3… mn ways respectively and T will be completed by completing each of these subtasks, then the number of ways of completing T will be m1 x m2 x m3 x ……. x mn”
I’ll stop here, and continue with The Addition Principle in the next part.