SAJC Check your Understanding_P&C_solutions

Check your Understanding (P&C) Section 1: Combinations 1) A committee of five members is to be chosen from seven men and five women. Find the number of ways which this committee can be formed if (i) the committee consists of three men and two women, [2] (ii) the man Aaron or the woman Beatrice or both of them must be in the committee. [3] (i) Number of ways 75 32CC 35 10 350    (ii) Method 1 (using complement) Total number of ways = Number of ways of forming committee – Number of ways of forming committee without Aaron and Beatrice 12 10 55CC 792 252 540    Method 2 Case 1 – Aaron is in the committee Number of ways 11 4C 330 Case 2 – Beatrice is in the committee Number of ways 11 4C 330 Number of ways 10 3330 330 C   660 120 540   Method 3 Case 1 – Only Alan is in the committee Number of ways 10 4C 210 Case 2 – Only Beatrice is in the committee Number of ways 10 4C 210 Number of ways 10 3210 210 C   420 120 540  

2) Five men and five women took part in a competition. Of these ten people, three prize winners are chosen at random. Find the number of ways that the prize winners can be chosen such that there are (i) no restrictions, [1] (ii) at least two women. [2] (i) No. of ways to choose 3 prize winners without restriction = 10 3 120C  (ii) No. of ways all three winners are women = 5 3 10C  No. of ways two women and one man = 55 21 50CC Number of ways required = 60 3) A group of 18 students are to be selected from 12 boys and 15 girls to go for an Overseas Community Service Project. Find the number of ways in which the group can be formed if (i) an equal number of boys and girls is selected, [2] (ii) a particular boy and a particular girl cannot be selected together, [2] (iii) no more than 5 boys are selected. [3] 3(i) No of ways for 9 boys and 9 girls selected 12 15 99 1101100 CC  (ii) Required no of ways 27 25 18 16 2643850 CC  (iii) Case 1: 3 boys and 15 girls No of ways = 12 15 3 15 220CC Case 2: 4 boys and 14 girls No of ways = 12 15 4 14 7425CC Case 3: 5 boys and 13 girls No of ways = 12 15 5 13 83160CC Total of ways = 220 +7425+83160 = 90805

4) A school is required to send a delegation of 10 teachers to attend the Teachers’ Conference 2018. This group of 10 teachers are to be selected from a pool of 7 Mathematics teachers, 5 Humanities teachers and 4 Science teachers. (i) How many different delegations can be formed? [1] One of the Mathematics teachers is the brother of a particular Humanities teacher. (ii) How many different delegations can be formed such that the siblings cannot be in the same delegation? [2] 4 (i) No of different delegations 16 10 8008C (ii) Method 1 No of different delegations 14 14 9 10 2 5005 CC