Lecture 2. Combinatorics 2.1 DOUBLING THE LAW Theorem: [Legal doubling] If the first operation can be done in n1 ways and each way can be followed by a second operation that can be performed by n2 ways, and so on until a series of k operation, then the operation can be carried out jointly with n1 × n2 × n3 × … × nk ways Operasi-1 Operasi-2 ………. Operasi-k n1 cara n2 cara nk cara n1 × n2 × n3 × … × nk cara Example: 1. Suppose that in a matter of five types of test available? True False. How many ways about the fifth? can be answered. 2. Provided three points, namely 4, 5 and 6. Decide? how many tens of numbers that can be? formed, if each number can only be used? one time.

Theorem: [summation law] 2.2 LEGAL ADDITIVE Theorem: [summation law] If an operation can be completed with k alternative, the first alternative can be done in n1 ways, the second alternative can be done in n2 ways, and so on up to the alternative-k with nk ways, then the operation can be done with n1 + n2 + n3 + … + nk ways Operasi ………. Alternatif-1 Alternatif-2 Alternatif-k n1 cara n2 cara nk cara n1 + n2 + n3 + … + nk ways Example: Suppose that in a test provided three types of True-False questions. Of the three questions quite done 2 a matter of course. There are two ways how the question can be answered? 2.3 Permutations Definition: [Permutations] Permutation is an arrangement that can be formed from a set of selected objects partially or completely. Note: 1. The order noted in the permutation. 2. Permutation is a special form of legal multiplication

P(n,n) = n! = n × (n-1) × (n-2) × … × 2× 1. Theorem: If there are n different objects, then the number of different arrangements (permutations) of n objects is P(n,n) = n! = n × (n-1) × (n-2) × … × 2× 1. Tempat-1 Tempat-2 ………. Tempat-n n cara n-1 cara 1 cara n × (n-1) × (n-2) × …× 2 × 1 cara Note: 1. P(n,n) read permutations level n of n. 2. 0! = 1. Theorem: If similar objects are not distinguished, the number of permutations of n objects, the objects have the kind of first n1, n2 objects have the second type, and so on until the object has a type nk to k is with n1+ n2 + … + nk = n. Theorem: The number of permutations of n different objects, if the objects taken r at a time is Note: P(n,r) read rate r permutation of n.

Evidence: Tempat-1 Tempat-2 ………. Tempat-r n cara n-1 cara n - r + 1 cara n × (n-1) × (n-2) × … × (n-r+1) cara Example: 1. There are 3 math books, four books on physics? and 5 chemistry books. These books will be? arranged lengthwise in a bookcase. Determine? many possible ways:        a. if all the books are distinguished,        b. if all books should be grouped and differentiated,        c. if all books should be grouped and differentiated,? but the group of math books should be placed? The earliest,        d. if the same book is not distinguished (book? mathematics and physics are all different, while the? book consists of three books chemistry Chemistry I and two books? Chemistry II).    2. Four students will form a musical group        with musical instruments: guitar, piano, drums and keyboards. there is        how the formation of musical groups that can be formed:        a. if each student can play four instruments? The music,        b. if a student can only play the piano or guitar,           while his three friends can play all           musical instruments,        c. if student A and B can only play the piano or? guitar, while the other two can play? all musical instruments.

considered as a circular permutation only. 2.4 Permutation CIRCULAR Definition: [circular permutation] Circular permutation is a circular arrangement that can be made from a collection of objects taken in whole or in part. Note: 1. only consider circular permutation? differences in the relative position of an object with the object? on the right and left. 2. The arrangement follows a circular permutation of ABC, BCA? and CAB is the same. C B A C B A A B C 3. If the objects to symbolize xi-i, then n permutation            following x1 x2 x3 … xn-2 xn-1 xn x2 x3 x4 … xn-1 xn x1 x3 x4 x5 … xn x1 x2  xn x1 x2 … xn-3 xn-2 xn-1 considered as a circular permutation only.

Theorem: The number of circular permutations of n distinct objects is Example: There are 15 blocks consisting of: 6 beams? white, 4 red beam, 3-beam and two beams of blue black.? The beams will be arranged in a circle.? Determine the number of ways:      a. if all beams are distinguished.      b. if the beam color and must be differentiated? grouped.      c. if the beam color and must be differentiated? grouped, and the group must be white? adjacent to the red group. 2.5 Combinations Definition: [Combinations] The combination is a group that can be formed from a set of selected objects partially or completely. Note: the order is not considered in combination theorem: The number of combinations of n different objects when r objects are selected:

Theorem: The number of ways to divide n different objects into k cells, where the first cell has a capacity of objects n1, n2 second cell has a capacity of objects, and so on until all cells have the capacity nk k objects, as well as the order of the objects in each cell is not considered is: in which n1 + n2 +…+ nk = n. Evidence: n benda Sel-1 Kapasitas n1 Sel-2 Kapasitas n2 Sel-k Kapasitas nk ………. cara cara cara cara Theorem: If there are n objects will be assigned to k places, then there are as many as k^n how its spread. Note: In the above theorem there is no limit to the capacity of the place.

Example: 1. In Maths, students are required Example:    1. In Maths, students are required? answer and chose 10 of the 12 questions given.? Define: a. many different possible choices. b. many different possible options if the 3 problems? The first must be answered. c. many different possible choices if? at least 3 of the first 5 questions to be answered. 2. There are 3 teachers and 12 students who will go sightseeing with 3 cars. The first car, a second? and third each with a capacity of 4, 6 and 5? people. Determine the number of ways to: a. allocated to the 15 people in the car. b. allocated to the 15 people in the car,? if in every car should be a teacher. c. allocated to the 15 people in the car,? if student A and B must be in a car.    3. Three marbles will be deployed to the two boxes. a. Determine the number of possible deployment. b. Determine the number of possible deployment, if? each box must be filled at least 1 marbles.    4. Four of the teachers will be placed in two schools.        a. Determine how many ways a placement.        b. Determine how many ways the placement, if any? school gets at least one teacher.        c. Determine how many ways a placement, if each? each school gets 2 teachers.