2022 Stat 1 Permutations and Combinations -APQ Solutions (RVHS)

H2 Mathematics (9758) River Valley High School, Mathematics Department, 2022 1 Statistics Tutorial 1: Permutations and Combinations Additional Practice Questions 1. (a) In how many ways can five copies of a book be distributed among ten people if no one gets more than one copy? (b) In how many ways can five different books be distributed among ten people if each person can get any number of books? (a) Assuming the books are identical, this is equivalent of choosing 5 people from 10 who get a book. (10 5 ) = 252 (b) Each book has 10 choices of person it could go to. 105 = 100 000 2. (a) How many squares are there in the grid below: (b) How many squares are there in the grid below: (a) Split into cases. 1x1 squares: 3x3 2x2 squares: 2x2 3x3 squares: 1x1 Total number of squares = 14 (b) Consider: Number of squares in 4x5 block – number of squares involving bottom right tile. Number of squares in a 4x5 block: 1x1 squares: 4x5

H2 Mathematics (9758) River Valley High School, Mathematics Department, 2022 2 2x2 squares: 3x4 3x3 squares: 2x3 4x4 squares: 1x2 Squares using bottom right tile = 1 (1x1), 1 (2x2), 1 (3x3), 1 (4x4) Number of squares = 40 – 4 = 36 3. SRJC/2012/H2 Prelim Paper II/Q6 (a) Find the number of ways in which 6 different pens can be distributed among 5 students such that each student can receive any number of pens. [1] (a) Number of ways = 65 = 15625 (b) A fleet of 9 different cars are to be parked at a reserved bay of 9 identical lots of which 4 lots are on one side and 5 lots are on the other. Find the number of possible parking arrangements of these 9 cars (i) if two particular cars cannot be parked side by side. [3] (b)(i) Number of parking arrangements     9! cases where the two cars are taking th e side with 4 lots cases where the two cars are taking the side with 5 lots   77 239! 3!2!5! 4!2!4!CC   = 292320 (ii) if two particular cars can only be parked at corner lots. [2] (ii) Number of parking arrangements 4 2(2!)(7!)C = 60480

H2 Mathematics (9758) River Valley High School, Mathematics Department, 2022 3 4. ACJC/2012/Prelim Paper II/Q6 Four boys and three girls are at a playground. (a) In one of their games, all seven of them have to stand in a straight line. (i) If all the girls are separated, find the number of ways in which this can be done. [2] 4! × 5P3 = 24 × 60 = 1440 Or 4! × (5C3 × 3!) = 24 × 60 = 1440 (ii) If all the boys are to be together, find the number of ways in which this can be done. [2] 4! × 4! = 576 (b) In another game, only six of them can play at a time and the six players have to stand in a circle with all the girls separated. Find the number of ways in which this can be done. [3] Case (i): 3 boys and 3 girls 4C3 × (3  1)! × 3! = 4 × 2 × 6 = 48 Case (ii): 4 boys and 2 girls (4  1)! × 3C2 × 4P2 = 6 × 3 × 12 = 216 Or (4  1)! × 3C2 × 4C2 × 2! = 216 Total number of arrangements = 48 + 216 = 264 B B B G G G B B B G B G

H2 Mathematics (9758) River Valley High School, Mathematics Department, 2022 4 5. MI/2012/Prelim Paper II/Q5 (a) An interior designer is designing a tile pattern of 3 blue, 3 red, 3 green, 1 yellow and 1 purple tiles in a line. All the tiles are identical except for the colour. Find the number of different possible ways to arrange the tiles if (i) the green, yellow and purple tiles must be placed next to each other, [3] Consider the green, yellow and purple tiles as 1 unit. Number of ways to arrange the green, yellow and purple tiles within the unit = 5! 3! Arranging the unit with the rest of the tiles = 7! 3!3! Thus number of possible arrangements for the tiles = 5! 7! 3! 3!3! =2800 (ii) no blue tiles are placed next to each other, [3] Number of ways to arrange the 3 red, 3 green, 1 yellow and 1 purple tile = 8! 3!3! Number of ways to slot in the blue tiles = 9 3   Number of possible arrangements such that no blue tiles are placed next to another = 8! 3!3! 9 3   =94080 (iii) a red tile to be placed at the beginning and at the end of the line. [3] Number of possible arrangements such that a red tile at the beginning and another red tile at the end of the line = (3 3 3)! 3!3!  = 10080

H2 Mathematics (9758) River Valley High School, Mathematics Department, 2022 5 (b) A group of 10 people consists of 9 men and 1 woman. Find the number of ways which the group can be seated at a round table with identical chairs if (i) there is no restriction, [1] Number of ways =  10 1 ! 362880 (ii) 2 particular men, Caleb and James, do not want to sit beside the woman, but will like to sit together. [3] Number of ways   7 27 1 ! 2! 2! 60480C      The chairs at the table are replaced with 10 chairs of different colours. (iii) Find the number of ways which the group can be seated at a round table if Caleb and James must sti ll sit together, but they need not be separated from the woman. [3] Number of ways  9 1 ! 10 2! 806400     6. JJC/2012/Prelim Paper II/Q6 Find the number of ways in which 4 -letter code -words can be obtained from the word ENDANGERED if (i) there are no repeated letters, [1] ENDANGERED: 3E, 2N, 2D, 1A, 1G, 1R No of 4-letter code-words from E, N, D, A, G, R 6 4 4! 360 C  (ii) there are three “E’s” [2] Select 1 letter from N, D, A, G, R. No of 4-letter code-words with the chosen letter and 3 “E”s 5 1 4! 3!C = 20 (iii) there is at least one repeated letter. [3] Case I: 3 same letters

H2 Mathematics (9758) River Valley High School, Mathematics Department, 2022 6 No. of such code-words = 20 Case II: 1 pair of same letters No. of such code-words 35 12 4! 3602!CC    Case III: 2 pairs of same letters No. of such code-words 3 2 4! 182!2!C   No. of 4-letter code-words that contain at least 1 repeated letter 20 360 18 398    7. NYJC/2013/H2 Prelim Paper II/Q9 Find the number of ways in which the letters of the word SYSTEMATIC can be arranged if (i) there are no restrictions, [1] (ii) the two ‘T’s must not be next to each other, [2] (iii) there must be exactly 3 letters between the two ‘T’s, [2] (iv) the first letter is ‘Y’ and the last letter is a consonant. [3] Solution: (i) 10! 9072002!2! (ii) Method 1 (Complementation) No of ways No of ways the two Ts without restriction are next to each othe r            9!907200 2! 725760 Method 2 (Insertion method) 98! 72576022!  (iii) X X X X X X X X T T T T T T T T T T T T (Insertion method) 8! 6 1209602!

H2 Mathematics (9758) River Valley High School, Mathematics Department, 2022 7 (iv) Case 1: Last letter is S or T No of ways 8!2 403202!   Case 2: Last letter is M or C No of ways 8!2 201602!2!   Total number of ways 60480 Alternative method Y _ _ _ _ _ _ _ _ _ No of ways 6 8! 604801 2!2!    8. TJC/2013/H2 Prelim Paper II/Q7 (a) (i) At a wedding dinner, 6 men and 4 women are to be seated at a round table with 10 identical seats. Find the number of different arrangements if there are no restriction. [1]