10. uu. 12, 13. 14, 15. A motorist travels through eight sets of traffic lights, each of which is red or green. He is forced to stop at three sets of lights. (a) In how many ways could this happen? (b) What other number of red lights would give an identical answer to part (a)? Find how many ways the letters of the word SOCKS can be arranged in a line: (a) without restriction, (b) so that the two $s are together, (c) so that the two $s are separated by at least one other letter, (a) so that the K is somewhere to the left of the C (a) Pind the number of arrangements of the letters in SLOOPS iff (i) there are no restrictions, (iii) the two Os are to be separated, (ii) the two Os are together, (iv) the Os are together and the Ss are together. (b) How many arrangements of the letters in TATTOO are there, if the two Os are to be separated? Find how many ways the letters of the word DECISIONS ean be arranged: (a) without restriction, (b) 0 that the vowels and consonants alternate, (c) so that the vowels come first followed by the consonants, (d) s0 that the N is somewhere to the right of the D. In how many ways can the letters of the word PROPORTION ALITY be arranged so that the vowels and consonants still occupy the sarne places? A form has ten questions in order, each of which requires the answer ‘Yes’ or ‘No’ Find the number of ways the form can be filled in: (a) without restriction, (£) if'an odd number of answers are ‘Yes’, (b) if the first and last answers are ‘Yes’, () if exactly three answers are “Yes’, and (c) if two are ‘Yes’ and eight are ‘No’, they occur together, (4) if five are ‘Yes’ and five are ‘No’, (h) if the first and last answers are ‘Yes? and (e) if more than seven answers are “Yes exactly Four more are *Yes Containers are identified by a row of coloured dots on their lids, The colours used are yellow, green and purple. In any arrangement, there are to be no more than three yellow dots, no more than two green dots and no more than one purple dot, (a) If six dots are used, what is the number of possible codes? (b) What is the number of different codes possible if only five dots are used? (a) How many five-letter words can be formed by using the letters of the word STRESS? (b) How many five-letter words can be formed by using the letters of the word BANANA Six-letter words are formed using two As and four Bs. By referring to the relevant section, of the notes, show how all the different possible words can be derived from the expansion of (A+ BY. Find how many arrangements of the letters of the word TRANSITION are possible if: {a) there are no restrictions, (d) the Ns oceupy the end positions (b) the Is are together, (e) an N occupies the first but not the last position (©) the Is, Ns and ‘Ts are together, (f) the letter N is not at either end.16. 1. 18. 19. 20. Ten coloured marbles are placed in a row. (a) If they are all of different colours, how many arrangements are possible? (b) What patterns? [Hir; This will need a guess-and-check approach | the minimum aumber of colours needed to guarantee at least 10000 different EXTENSION Bob is about to hang his cight shirts in the wardrobe. He has four different styles of shirt, two identical ones of each particular style. How many different arrangements are possible if no two identical shirts are next to one another? Find how many arrangements there are of the letters of the word GUMTREE if: (a) there are no restrictions, (b) the Fs are together, (c) the Es are separated by: (i) one letter, (ii) two letters, (iii) three letters, (iv) four letters, (v) five letters, (a) the G is somewhere between the two Es, (c) the M is somewhere to the left of both Es and the U is somewhere between them, (£) the G is somewhere to the left of the U and the M is somewhere to the right of the U. If the letters of the word GUMTREE and the letters of the word KOALA are combined and arranged into a single twelve-letter word, in how many of these arrangements do the letters of KOALA appear in their correct order, but not necessarily together? Os Binary numbers consist of a sequence of Os and 1s. One such number contains exactly and at most b 1s. (a) If the number eon! a+b digits, show that there are “°C; possible sequences. You Aaa (b) Prove that: Cot OFC) + 48Cy pore HM (c) Hence show that the total number of pessible sequences is will need to reall that "C, = HEHIC, HG,Exercise 10G — 1 Two people are chosen from a group of five people called P, Q, R, S and ‘List all possible combinations, and find how many there are . Find how many ways you can form a group of: (a) two people from a choice of seven, (©) two people from a choice of six, (b) three people from a choice of seven, _(d) five people from a choice of nine, . (a) Find how many possible combinations there are if, from a group of ten people: (i) two people are chosen, (ii) eight people are chosen. (b) Why are the answers identical? From a party of twelve men and eight women, find how many groups there are of: (a) five men and three women, (b) four women and four Four numbers are to be selected from the set of the first eight positive integers. Find how many possible combinations there are if: (a) there are no restrictions, (c) there is exactly one odd number, (b) there are two odd numbers and two (4) all the n even numbers, (e) there is at least one odd number. bers must be even, Four balls are simultaneously drawn from a bag containing three green and six blue mar bles. Find how many drawings are possible if: {a) the balls may be of any colour, (¢) there are at least two green balls, (b) there are exactly two green balls, (d) there are more blue balls than green balls A committee of five is to be chosen from six men and eight women. Find how many committees are possible if (a) there are no restrictions, {e) there are four women and one man, (b) all members are to be female, (f) there is a majority of women, (c) all members are to be male, (g) a particular man must be included, (4) there are exactly two men, (h) a particular man must not be included.6a) 56 (b) 5 7la) 60 (b) 24 (c) 36 @) 30 (half of them) Bay) 180 qi) GO (ii) 120 iy 24 (b) 40 ga) 90720 (b} 720 (ce) 720d) 45360 (half of them) 10 2721600 tia) 1024 (b) 256 (ce) 45 (d) 252 @) 56 f 512 @) 8 (hy 70 12(a) 60 (b) 60 13(a) 120 (b) 60 15(a) 453600 = (b) 90720 =e) 5040 -~— a): 10080 (@) 80640 (f 282240 16(a) 3628800 (b) 4 17 864 18(a) 2520) 720 ey) 600 =i) 480 Git) 360 (iy 240 (w 120 (a) 840 € 210 (fh 420 19 1995840Exercise 10G (Page 430) 1 There are ®C2 = 10 possible combinations: PQ, PR, PS, PT, QR, QS, QT, RS, RT and ST. aa) 21 ) 35 @ 15 @ 126 stay) 45 il) 45 ©) 2°C, = Cy, and in general "C, ="Cp_y. 4a) 44352 (b) 34650 5(a) 70 b) 36 (9 16 1 (@ 69 6a) 126 ) 45 ©) 51 ad) 75 Ta) 2002 (b) 56 () 6 @) 840 ¢e) 420 (f 1316 @ 715 (hy) 1287