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.