Counting began much earlier. We can trace it to ancient civilizations. In fact, counting began much earlier than the onset of ancient civilizations. Counting of objects was felt necessary even in the most primitive human society. Humans wanted to know the number of animals or the number of fellow humans. Early humans had some idea of counting but it took a long time to invent symbols for counting.
In modern human society, we teach our children mathematics with counting of numbers. It is an important elementary training that is necessary to comprehend the subject of mathematics. The first operator we use is addition to count numbers. Then, slowly we introduce multiplication to count the numbers fast. For example, if there are five sets of objects and each set contains five objects, then we use multiplication operator to find the total number of objects instead of the long process of counting each object using the simple addition operator.
We did not stop there. We invented advanced models like permutation and combination that simplify our counting process. For example, if a person can go from city A to city B in 4 distinct routes and from city B to city C in 3 distinct routs, then the person can take 12 distinct routes to reach city C from city A, provided there is no other direct route from city A to city C. We just multiply 4 and 3, that is, you have to choose one route from A to B and another route from B to C. There are four ways to choose from the former and three ways to choose from the latter. 4 and 3 or 4×3 or 4C1×3C1. The letter ‘C’ indicates combination, the rule for selection. 4C1 means choose 1 from 4. If there are 4 distinct routes you can choose a route in 4 ways.
What is fascinating is how our mind evolved to count in different ways to arrive at the same result. At one point in the evolution, humans were unable to quantify the objects though had some idea about quantity. This is not far away in the evolutionary time scale. Invention of number symbols especially a symbol for zero ought to be the inflection point for the growth and development of counting in mathematics. Instead of manually counting objects we are using principles of counting in order to save our time and energy, which had probably driven our mind to invent computing machines.
How many arrangements can be made from the letters A, B, C, and D, using all letters without repetition?
ABCD, ABDC, ACBD, ACDB, ADBC, ADCB; BCDA, BCAD, BDAC, BDCA, BACD, BADC; CABD, CADB, CBAD, CBDA, CDAB, CDBA; DABC, DACB, DBAC, DBCA, DCAB, DCBA.
This is how we manually write down all possible words and count them to arrive at the answer. It’s tedious and time-consuming. Imagine, how much time will it take if you are given all the 26 letters?
Let us see how our mind had evolved to count the number of arrangements without manually writing down all the possibilities. Look at the data. I have written all the possible arrangements systematically in order not to allow repetition. This is the first step. There are four letters and therefore we need four places to write each letter. There are four sets of arrangements and each set contains six arrangements. Hence, we have 4×6=24 arrangements. The first set starts with the letter ‘A’, the second set starts with the letter ‘B’, and so on.
Now, let us think in a different way. There are four places and there are four letters to occupy those four places. In the first place, any of the four letters can occupy in four ways, that is A, B, C, or D. In the second place, we have only three letters since we have already consumed one letter for the first place. Similarly, for the third place we have two letters and for the fourth place we have only one letter. Therefore, the number of arrangements will be 4×3×2×1=24. Symbolically, we represent it as 4! (Read as 4 factorial) or 4P4 (P represents permutation, the rule for arrangements).
There is also another explanation for this result. We can use the combination. There are four letters and four places. We are going to place one letter in the first place. Since there are four letters we can choose in 4C1 ways. For the second place, since we are left with only three letters, we can choose in 3C1 ways, and so on. Hence, we have 4C1×3C1×2C1×1C1=24.
Following the principles of counting, we can now count arrangements easily even if you are given all the 26 English alphabets. This is the power of our mind. It’s this power of mind that led us to invent many things. To really appreciate our power of mind I would like to give you another illustration.
5 letters are to be sent to 5 different addresses. The 5 addresses are written on 5 different envelopes. In how many ways can all the 5 letters go to the wrong envelopes?
This particular problem is solved using the principle of inclusion and exclusion, viz. derangements (symbolically represented as !n)
The problem of counting derangements was first considered by Pierre Raymond de Montmort in 1708. He solved it in 1713, as did by Nicholas Bernoulli at about the same time (Wikipedia). The derangement formula is: .
Now, just substitute n=5 in the above formula, you will get the answer as 44. We have to do a bit of research to find how Pierre conceived this idea and arrived at the above formula. However, I had tried to do a bit of research on how the human mind could have evolved to find this formula. Sitting on the shoulders of wise men, I first used the induction method. If there is one letter and one envelope, there is zero way of putting in the wrong envelope. Look at the formula, the first two terms cancel it out. It gives you the answer as zero.
If there are two letters and two envelopes, you can place the two letters in only one wrong way. Look at the formula, the first three terms give you the answer as 1. I continued the induction method up to 5 letters and found the answer as 44. It took a lot of time to do it manually. However, I determined to do it to find the pattern. Pattern finding is my passion. The first 5 letters gave me a sequence like 0, 1, 2, 9, and 44. Five results are a safe bet to find a generating term as mathematicians suggest. I tried the method of differences. I could not find a meaningful pattern. I really got frustrated at one point of time. I looked back at my manual workings for each case. I split the numbers in various combinations to arrive at a pattern. Finally, I got the pattern. You have to combine the previous two terms and multiply with an integer to get the next number.
(2+9)×4=44 and so on.
Given the first two results, which you can easily manually find, you can generate the subsequent terms. All you need is 0 and 1. I checked for the next numbers and found that they are obeying the pattern.
So, the generating term I got was: Sn = (Sn-2+Sn-1)×n, n≥3; S1=0, S2=1.
The above derangement question was asked to me by a student in 2015. It’s at that time that I used the induction method to find the answer. At that time, I had not come across the derangement formula. That was a blessing in disguise, otherwise I would not have gone for induction method.
In 2016, the same question was again asked to me by a student. It’s at that time, that I tried to find a pattern and got it too.
Surprisingly, in 2017, when I looked back at my workings, I found another model to find the same result. This time it was even much easier and simpler. 5 letters and 5 envelopes. 5 distinct things in 5 distinct places. The total arrangements are 5!=120, which we have already worked out in the earlier example. What we have to do is divide n! successively with divisors starting from 2, 3, 4, and so on up to n. In our case, n = 5. Then, find the sum of the sequence of all quotients with operators minus and plus successively as used in the derangement formula.
Therefore, the formula according to my finding is:
n!/2 – Q1/3 + Q2/4 – Q3/5+…+ (-1)^n*Qn-2/n; Q1, Q2,… are successive quotients.
Our mind is designed to evolve. Should we stop our mind evolving? What are we teaching to our children? Most of the millennial do not know multiplication and division. The few ones who can do division hate finding three decimals for an accurate answer. It’s a dangerous trend that will drive our children’s mind to that of primitive society. Are we pushing our younger generation back to primitive age? Are we deliberately setting a stage for a different human species to branch out? It’s not improbable for the humans to branch out into a new species in the scheme of evolution. However, that is the work of evolution. Or, the new species is already existing and destroying the not so smart human species. The millennial are a bit lazy and depend too much on machines to do their works. In the process, their mind is becoming dormant and not evolving to keep up with others (the few smart children). Let us do something before it is out of our hand.