Permutations and Combinations IIT JEE+Olympiad Lecture 1
- 2. Guidelines 1.Try not to disturb or distract your classmates. 2.Discuss freely about in-class topics or solutions during problem solving sessions. 3.Raise your hand to ask questions during class. 4.Feel free to ask any class-related doubts after class. PnC-1 2
- 3. Motivation • Calculation of the number of desired/undesired outcomes(events) • Using the above, forecasting the probability of those outcomes • Real Life Application ->Decryption in Cryptography, Puzzle solving,Hacking ->Algorithms : Finding out (and improving on) the speed(runtime) of any program/process PnC-1 3
- 4. PnC - Learning Goal • Understanding ‘Counting’ as an application of logical operations • Accounting for “Double Counting” of Identical per./com. for correct calculation of nPr, nCr (Distinguishable objects)(Lecture 1,2,3) • Finding the number of ways of Partitioning Distinguishable and Indistinguishable Objects(Lecture 3,4) • Applying Inclusion-Exclusion Principle through Venn Diagrams(Lecture 4) PnC-1 4
- 5. PnC-1 Topics 1.Principles of Counting + Application to Graphs 2.Counting the Number of Sequences 3.Permutations + Calculation for Distinguishable Objects 3.1.Dealing with Indistinguishable Objects PnC-1 5
- 6. 1.The Two Principles of Counting A)Principle of Addition: “If the number of ways of doing ‘X’ is ‘m’ and the number of ways of doing ‘Y’ is ‘n’, then the number if ways of doing either of ‘X’ OR ‘Y’ them is ‘m+n’’ B)Principle of Multiplication: “The number if ways of doing both of ‘X’ AND ‘Y’ them is ‘m×n’” Note: Order of doing ‘X’, ‘Y’ in terms of time is not important if the events do not depend on the order of doing them Example: For real numbers ab=ba; but for matrices AB≠BA => dependent on order PnC-1 6
- 7. 1.Simple Counting in Graphs A)Principle of Addition in action: 1.Number of ways of choosing a one digit number OR a 2 digit number = 10+90 = 100 2.Number of Paths that lead you out of B = ? 3.Number of ways of choosing a one digit number OR a one digit number = ? B)Principle of Multiplication in action: 1.Number of ways of choosing any 2 one digit numbers = 10×10 = 100 2.Number of paths from A to C = A to B AND B to C = 6×4 PnC-1 7
- 8. 1.1.Relation to Basic Logical Operations Addition relates to XOR (Exclusive OR) Mutiplication relates to AND ANDOrder of R1,R2 Doesn’t matter. They could even be merged into (R1+R2) XORAn electron flows through either but not both.Which branch gets which resistor doesn’t matter PnC-1 8
- 9. Problem Session 1(5 min.) Q1) 1. Number of paths from A to C = ? 2. Number of paths that lead you to C = ? 3. Number of ways to get out of A = ? Q2)Number of ways to factor ‘20’ into two Natural numbers = ? PnC-1 9
- 10. 2.Counting the Number of Sequences Consider a sequence of Events E1,E2,…En that are causally independent ->Irrespective of the order of choosing, total number of sequences possible= (No. of choices for E1) ×(….…× (No. of choices for En) Example1) The Hermetian alphabet consists of only three letters: A, B, and C. A word in this language is an arbitrary sequence of no more than four letters. How many words does the Hermetian language contain? ->Hint: Ans = number of 4 letter words + 3 letter words +……. ->Consideration : What if you cannot repeat the letters in a word? PnC-1 10
- 11. 2.Counting the Number of Sequences Example2) Elections were held in the Indian Cricket team to select a Captain (E1) and a Vice Captain(E2). How many outcomes are possible? Events E1,E2 where E2 is causally dependent on E1 ->Now we must start by choosing for the independent event E1. Total number of sequences possible=(choice 1 for E1)×(choices left for E2 in this case) + (choice 2 for E1)×(choices left for E2 now) + …….. ->Crosscheck this with Q1) in session 1 If choices left for E2 are same in all cases, number of sequences possible=? PnC-1 11
- 12. 3.Permutations – Stacking the ORs! Events E1,E2,…En where Ei+1 is causally dependent on Ei Assumption for Permutations: Number of choices left for every Event is the same regardless of choice of last Event ->Start by choosing for the independent event E1. Total number of sequences possible=(choices for E1)*(choices left for E2)*(choices left for E2)×……×(choices left for En) Example)Number of ways to order 4 (distinct) Pizza items on the phone = 4×3×2×1 = 4 Permutation: An assignment of Position numbers from 1 to ‘n’ to ‘n’ objects for any natural number ‘n’. PnC-1 12
- 13. 3.Calculation of Permutations(nPn) For Distinguishable Objects O1,O2,…On , we want to find total number of sequences with no repetitions. (a One-One Onto mapping in calculus) Event Ei = Giving object Oi a position (from 1 to n) Distinguishable Set of Objects: Interchanging any 2 Objects gives a different sequence Answer) Assumption is satisfied => n×(n-1)….×2×1 = n! := nPn Oral1) Number of sequences O1,O2,…On with repetitions allowed = ? Oral2) Is 96! Divisible by 97? PnC-1 13
- 14. 3.1.Some objects Indistinguishable Indistinguishable Objects : 2 Objects are identical when they give the same sequence even after interchanging Example1)Number of ways to order 4 Pizza items on the phone, two of which are identical ≠ 4×3×2×1 Why? Since we need to account for ‘Double Counting’ of equivalent sequences. Notice that in doing 4×3×2×1, each sequence is counted twice. Oral4) Ways of Ordering ‘m’ pizza items, ‘k’ of which are identical = ? PnC-1 14
- 15. 3.1.Some objects Indistinguishable Oral4) Ways of Ordering ‘m’ pizza items, ‘k’ of which are identical = ? Answer) We could permute those ‘k’ objects anyhow to get the same i.e. one valid sequence. Treating all objects as distinguishable would count each valid sequence k! times. Hence, answer = m!/k!. Example2) Elections were held in the Indian Cricket team to select two Captains from 11 players. How many outcomes are possible? Hint) First look back to the earlier example, and notice the difference in the distinctness of the titles. This is the build-up to number of combinations nCr read as n ‘choose’ r. PnC-1 15
- 16. Final Problem Session(15 min.) Q1) Ways of Ordering ‘m’ pizza items, ‘k1’ of which are the same pizza and ‘k2’ of which are the same beverage (m≥k1+k2) = ? Q2) Getting used to factorials : Given that (-1)!= , prove that Q3) 1.How many diagonals are there in a convex n-gon? 2.How many triangles are there inside this n-gon? Note: triangles may include edges of n-gon. PnC-1 16
- 17. Reference Material 1. CBSE 11th textbook for introductory material 2. Chapters 2 and 11 on Combinatorics of Mathematical Circles(Russian Experience), American Mathematical Society for JEE, especially the practice problems PnC-1 17