Permutations with Repetitions - In the world around us
As Kabani tried to read more about permutations with repetitions, she comes across several examples: Secret locks, Morse code, Genetic code etc.
One of them was that of secret locks (also called combination locks). These locks open up only when a specific combination of numbers or alphabets is dialed.* One example of such a combination lock is the ATM card pin number. Each pin number has 4 digits and each digit's place can be filled with any of the ten numbers (0, 1, 2... 9). Using the method we have seen earlier, the total number of calculations is found to be 104 = 10000. Which means that of these 10000 combinations, 9999 will be failures. (ATM cards usually stop working after typing the wrong pin number thrice, so even if someone gets your card there is very little chance that they can get the right number in three attempts. 9999 is too big you see!)
Morse Code
The Morse code is used in telegraph communications. In this code, the alphabets, numbers and punctuation marks are represented by dots and dashes. Some characters are represented by just a single sign (by a single sign we mean one (.) or (-)), like E (.), T (-); whereas others use five signs, like zero, 0 (- - - - -), 1 (. - - - -) etc. But why the number 5? The answer lies in the number of permutations with repetitions possible. Here, we can fill each place with either dot (.) or dash (-), i.e. 2 options. Suppose we are using only one sign, it is possible to have 21 (= 2) distinct arrangements and hence we can transmit only 2 letters (E and T). Using two signs, we can transmit 22 (= 4) letters, using three signs we can transmit 23 (= 8) letters and using four signs we can transmit 24 (= 16) letters. Thus the total number of letters we can transmit using up to four signs is 2 + 4 + 8 + 16 = 30 which will not be sufficient for transmitting all the alphabets, numbers and punctuation marks. Using five signs, we can additionally transmit 25 (= 32) letters and so in total we will now have 62 options which is quite sufficient. The Morse code described here makes no difference between alphabets written in the upper case and lower case. If we were to incorporate that, how many signs would we need?
Genetic code
Breaking the genetic code has been one of the most remarkable achievements of the twentieth century. Scientists now know how genetic information is passed on from one generation to the other. The genetic information is stored in giant molecules of deoxyribonucleic acid (DNA). Each molecule of DNA is an arrangement of the four nucleotides (adenine, thymine, guanine and cytosine). Molecules of DNA differ in the arrangement of these nucleotides and they determine the order in which the proteins are built from the 20 amino acids. Each amino acid is in the form of a code made up of three nucleotides. Why codes of only 3 nucleotides? The answer is similar to the one in Morse code. Using combinations of just one or two nucleotides would not result in sufficient number of combinations for the 20 amino acids and the START / STOP functions. Hence, codes of three nucleotides would be required. It is interesting to see how nature takes advantage of so much redundant information – the number of combinations is 64 while the number of amino acids is only one third! (Do find out more about how nature uses the excess of combinations in regulating the protein expression and fighting mutations).
A single chromosome contains millions of nucleotides. The number of distinct chromosomes possible is 4N , where N is the number of nucleotides in the chromosome. The number is just too big to write it down here, however only a very small fraction of these have been sufficient to ensure the diversity of all life on this planet. Why only a few? These are questions yet to be answered.
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