Combinatorics - 3
Kabani was making progress with her counting of how many possibilities are there in anything she did. Sometimes she found it easy like switching a light on or off, sometimes it became tougher like the possible lengths of a line she could draw with a 15cm scale. Her parents couldn't help her either on this. Her unsolved problem list was ever-increasing now. It was a big motivation to explore further...
Kabani observed that while calculating the number of possibilities sometimes she was adding and sometimes she was multiplying and she never did them together. Now this was strange! She wasn't told where to do what, then how did she know - when and what to do. Look at the following problems -
There are 5 apples and 3 oranges, the number of ways you can select an apple or an orange is ______.
There are 5 apples and 3 oranges, the number of ways you can select an apple and an orange is ______.
The two questions varied in just one word and that makes all the difference. When Kabani was given the choice to select an apple or an orange, she was selecting one object out of the 8 objects as there was no emphasis on what she should be selecting. So the number of ways she could have done that was 8. However, when she was told she had to select both an apple and an orange, the situation is very different. She can select any one of the 5 apples, so the selection is possible in 5 ways. Once she has selected the apple, she can select an orange in 3 ways. After every choice of an apple there are 3 further choices. So in all she can make a selection in 3 + 3 + 3 + 3 + 3 (=15) ways, i.e., 3*5 or 5*3 ways. (Do you remember the law which says 3*5 = 5*3?). What Kabani thought was multiplication was just repeated addition! But why repeated addition?
The reason lies in the fact that, in the first case Kabani was doing only one work but in the later she had two tasks. She could do the first in a few different ways and then later do the second in few more ways, thereby resulting in a huge number of ways she can complete both the tasks together.
Mathematicians call the first case as the law of sums and the second as the law of products. The names hardly matter... may be they should better be called as the law of addition and the law of repeated addition. It is for you to decide, what you would like to call them. How about Alpha Rule and Beta Rule? You are now one step ahead in your understanding of combinatorics, that is what matters.
Until next time -
A "sign" is being assigned to every person in a village. Each "sign" consists of a geometrical figure (triangle, square, rectangle or circle), an alphabet and a single-digit number. How many unique signs can be made?
NEXT (coming up)
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