Counting With Repetitions

Transcription

1 Counting With Repetitions The genetic code of an organism stored in DNA molecules consist of 4 nucleotides: Adenine, Cytosine, Guanine and Thymine. It is possible to sequence short strings of molecules. One way to sequence the nucleotides of a longer string of DNA is to split the string into shorter sequences. A C-enzyme will split a DNA-sequence at each C. This means that each fragment will end at a C except possibly the last fragment. Similarly for A-enzymes, G-enzymes and T-enzymes. If the original nucleotide is split on each of C, A, G and T then it can be sequenced as it is most likely a unique sequence that can be constructed by each of the four sets of fragments.

2 Example. Given a 20-nucleotide string split at the Cs, one might have the fragments: AC, AC, AAATC, C, C, C, TATA, TGGC Q. How many different 20-nucleotide strings could have given rise to the above set of fragments? In other words, how many different arrangements are there of these fragments? 2

3 Theorem. Given n objects, with r of type, r 2 of type 2,..., r m of type m where r + r r m = n then the number of arrangements of the n objects, denoted by P (n; r, r 2,..., r m ) is: = ( n r )( n r )( n r r ) ( 2 n r r 2 r ) m r 2 r 3 r m Q. What does this formula simplify to? 3

4 Proof. P (n; r, r 2,..., r m ) = n! r!r 2!... r m! Convert the problem into something we can already solve. Suppose all objects are distinct. How many permutations are there? Consider the i th type. It is repeated r i times. Add the subscripts, 2,..., r i to make them unique. Example. Let our set of objects be the letters {b, a, a, a, n, n}. We can make the as and ns unique as follows: a, a 2, a 3, n, n 2. Q. Number of permutations of {b, a, a 2, a 3, n, n 2 }? We can think of this as the number of ways to arrange {b, a, a, a, n, n}. Q. How do we say this mathematically? 6! =?? 4

5 Q. Suppose we have b, a, a, n, n, a. What are the different ways to arrange the as? Q. How many ways to arrange the ns for each arrangement of as? Resulting in: Generally, for the i th type, we have the objects. Therefore: n! = ways to arrange n! = Rearranging gives: P (n; r, r 2,..., r m ) = 5

6 Selections With Repetitions Example. While shopping at the St. Lawrence market, you decide to buy half a dozen bagels. There are three flavours to choose from. Q. How many different ways can you select your 6 bagels? Rephrase as an arrangement problem. Sesame Poppy Seed Plain xx xxx x Q. How is this an arrangement problem? Q. How can we also think of this as a selection problem? Let n be the number of choices, r the number of items selected. Then 8 = r + (n ) and we are choosing r = 6. This results in: 6

7 Theorem. Given r objects and n types of objects to choose from, the number of selections with repetitions is: ( r + (n ) ) Proof. Simply generalize the arguments above. r Example. How many ways are there to select a committee of 5 politicians from a room full of indistinguishable Democrats, indistinguishable Republicans and indistinguishable Independents if every party must have at least two members on the committee? Solution. 7

8 Pascal s Triangle Blaise Pascal [ ] was a French mathematician, physicist, inventor, writer and philosopher. As a teenager, he invented the mechanical calculator. He collaborated with Pierre de Fermat in Probability Theory influencing modern economics and social sciences. Invented Pascal s Triangle in his Treatise on the Arithmetic Triangle. Pascal s Triangle can be constructed several ways. One way is: The edges are s. The interior numbers are the sum of the two numbers above. 8

9 Fill in the entire triangle. Q. What do you notice about each number? How is it related to pizza and toppings? Q. How is Pascal s Triangle related to binomial expansion? I.e., how is it related to the coefficients of the polynomials found by expanding (a + b) n? Let s write out: (a + b) = (a + b) 2 = (a + b) 3 = (a + b) 4 = etc. 9

10 Fun Pascal Triangle Facts* Q. What do each line sum to? Q. Read each line as a number. If a number has two digits carry the tens digit to the left and add. What do these numbers represent? * 0

11 More Cool Pascal Triangle Facts Q. Colour all the odd numbers. What does this remind you of? A well known repeating pattern, the Sierpinkski Triangle.

12 Q. What series of numbers do you get by adding up the numbers of the same colour which are on a stretched diagonal? Q. What numbers do you get if go down the diagonals? ones counting numbers 2 triangular numbers 3 3 tetrahedral numbers

13 Q. What are the triangular numbers? They are the numbers you get by counting the number dots required to make successive triangles**. Guess what the Tetrahedron numbers are. ** 3