PnC Qns and Solns (SAJC)

Chapter 1: Permutation and Combination 1. HCI JC2 Prelim 8865/2019/Q6 (i) Janet is going on a holiday trip and is trying to recall the 4-digit numerical combination lock for her luggage. However, she can only remember the numerical combination consists of digits from the set { 0, 1, 4, 6, 8, 9} . (a) How many different numerical combinations are there if the odd and even digits must alternate and repetitions of digits are not allowed? [1] (b) If the 4-digit numerical combination must be a number that is more than 6400 and that there are no repetitions of digits, find the maximum number of numerical combinations Janet has to try in order to open her lock? [2] (ii) Janet has 6 T-shirts, 3 sweaters and 3 pairs of jeans, all of which are of different designs. Find the number of ways she can select her clothes for her trip if she needs 4 T -shirts, a sweater and at least a pair of jeans. [3] Answer: (ia) 48, (ib) 156, (ii) 315 6. HCI JC2 Prelim 8865/2019/Q6 (Solutions) (ia) Set of digits to choose from = {0, 1, 4, 6, 8, 9} Note: Zero is even. Case 1: odd even odd even = 2 4 1 3 24   = Case 2: even odd even odd = 4 2 3 1 24   = No. of combinations = ( )2 4 3 2 1 48    = (ib) Case 1: 6 _ _ _ 4 or 8 or 9 No. of combinations = 1 3 4 3 36   = Case 2: _ _ _ _ 8 or 9 No. of combinations = 2 5 4 3 120   = Total no. of combinations 36 120 156+= (ii) 6T, 3J, 3W No. of ways to select T-shirts = 6 154  =

No. of ways to select jeans = 333 71 2 3      ++=           No. of ways to select sweater = 3 Total no. of ways = 15 7 3 315  = 2. NJC JC2 Prelim 8865/2019/Q8 Find the number of ways in which the letters of the word QUESTION can be arranged, if (i) Q and U must be next to each other, [2] (ii) E and S must be separated, [2] (iii) consonants (Q, S, T, N) and vowels (U, E, I, O) must alternate. [2] A code consists of 5 letters chosen from {Q, U, E, S, T, I, O, N}. Repetitions are not allowed. (iv) Find the probability that a randomly chosen code has only 1 vowel. [2] Answer: (i) 10080, (ii) 30240, (iii) 1152, (iv) 1 14 2. NJC JC2 Prelim 8865/2019/Q8 (Solutions) (i) No. of ways to arrange Q and U 2!= No. of ways to arrange all letters with Q and U grouped together 7! 5040== No. of ways that Q and U must be next to each other 7!2! 10080== (ii) Method 1: Slotting No. of ways to arrange the other 6 letters 6! 720== No. of ways to choose slots and arrange E and S 7 2! 422 =  = No. of ways E and S must be separated 76! 2! 302402 =   = (ii) Method 2: Complementary No. of ways to arrange all the 8 letters 8! 40320== No. of ways E and S must be separated 8! (i) 30240= − = (iii) To alternate, the pattern must be CVCVCVCV or VCVCVCVC, where C = consonan