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school. “Paper 42” means Paper 4, variant 2.
Practice questions
Original questions written by HomiLabs in the style of the exam, with answers and mark schemes. They are not
Cambridge past paper questions and have not been reviewed by Cambridge. Try each one before opening the answer.
Q1Permutations · Core · Paper 1 style · no calculator · 3 marks
(a)Find the number of ways in which 7 different books can be arranged in a row on a shelf.[1]
(b)Two of the books are dictionaries. Find the number of arrangements in which a dictionary is at each end of the row.[2]
Show answer
(a) 5040
7! = 5040 B1
(b) 240
2! × 5! M1; 240 A1
Q2Permutations · Core · Paper 2 style · calculator · 3 marks
(a)A 4-digit code is made using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9. No digit may be used more than once. Find the number of possible codes.[1]
(b)Find the number of these codes that are even.[2]
Show answer
(a) 3024
⁹P₄ = 3024 B1
(b) 1344
last digit from {2, 4, 6, 8}: 4 × 8 × 7 × 6 M1; 1344 A1
Q3Permutations · Core · Paper 1 style · no calculator · 5 marks
(a)Find the number of different arrangements of the 6 letters of the word MULTAN.[1]
(b)Find the number of these arrangements that begin with M and end with N.[2]
(c)Find the number of arrangements in which the two vowels are next to each other.[2]
Show answer
(a) 720
6! = 720 B1
(b) 24
4! M1; 24 A1
(c) 240
5! × 2! M1; 240 A1
Q4Permutations · Core · Paper 2 style · calculator · 2 marks
(a)10 runners take part in a race. Find the number of ways in which the gold, silver and bronze medals can be awarded, assuming there are no ties.[2]
Show answer
(a) 720
¹⁰P₃ M1; 720 A1
Q5Permutations · Standard · Paper 2 style · calculator · 3 marks
(a)5-digit numbers are formed from the digits 0, 1, 2, 3, 4, 5, 6. No digit is used more than once and a number cannot begin with 0. Find how many of these numbers are greater than 40 000.[3]
Show answer
(a) 1080
first digit 4, 5 or 6: 3 ways B1; remaining 4 places from 6 digits: ⁶P₄ = 360 M1; 3 × 360 = 1080 A1
Q6Permutations · Standard · Paper 2 style · calculator · 4 marks
(a)4-digit numbers are formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, with no digit repeated. Find how many of these numbers are odd and greater than 5000.[4]
Show answer
(a) 420
first digit 5 or 7: 2 × 3 (odd last digits left) × 6 × 5 = 180 M1 A1; first digit 6 or 8: 2 × 4 × 6 × 5 = 240 A1; total 420 A1
Q7Permutations · Standard · Paper 2 style · calculator · 5 marks
(a)6 boys and 3 girls stand in a row for a school photograph. Find the number of arrangements in which the 3 girls stand next to each other.[2]
(b)Find the number of arrangements in which no two girls stand next to each other.[3]
Show answer
(a) 30 240
7! × 3! M1; 30 240 A1
(b) 151 200
boys 6! B1; girls into 7 gaps: ⁷P₃ = 210 M1; 720 × 210 = 151 200 A1
Q8Permutations · Standard · Paper 1 style · no calculator · 4 marks
(a)The 7 letters of the word FORMULA are all different. Find the number of arrangements in which the three vowels O, U and A are together.[2]
(b)Find the number of arrangements that begin and end with a consonant.[2]
Show answer
(a) 720
5! × 3! M1; 720 A1
(b) 1440
4 × 3 × 5! M1; 1440 A1
Q9Permutations · Standard · Paper 2 style · calculator · 5 marks
(a)An online password consists of 2 different capital letters (from A to Z) followed by 3 different digits (from 0 to 9). Find the number of possible passwords.[2]
(b)Find how many of these passwords start with a vowel (A, E, I, O, U) and end with an even digit.[3]
correct use of n!/(n − r)! M1; simplification to n − 2 A1
Q11Permutations · Challenge · Paper 2 style · calculator · 4 marks
(a)4-digit numbers are formed from the digits 0 to 9 with no digit repeated; a number may not begin with 0. Find how many of these numbers are divisible by 5.[4]
Show answer
(a) 952
ending in 0: 9 × 8 × 7 = 504 M1 A1; ending in 5: 8 × 8 × 7 = 448 M1; total 952 A1
Q12Permutations · Challenge · Paper 2 style · calculator · 5 marks
(a)At a conference, 3 Pakistani, 2 Chinese and 2 Turkish delegates sit in a row of 7 seats. Find the number of arrangements in which the delegates of each country sit together.[3]
(b)Find the number of arrangements in which a Pakistani delegate sits in each of the two end seats.[2]
Show answer
(a) 144
3! for the order of countries M1; × 3! × 2! × 2! M1; 144 A1
(b) 720
³P₂ × 5! M1; 720 A1
Q13Permutations · Challenge · Paper 1 style · no calculator · 4 marks
(a)3-digit numbers are formed from the digits 1, 2, 3, 4, 5, 6, with no digit repeated. Find how many of these numbers are even and greater than 300.[4]
Show answer
(a) 40
first digit 3 or 5: 2 × 3 × 4 = 24 M1 A1; first digit 4 or 6: 2 × 2 × 4 = 16 A1; total 40 A1
Q14Permutations · Challenge · Paper 2 style · calculator · 5 marks
(a)8 friends, including Ali and Bilal, sit in a row of 8 seats at a cinema. Find the number of arrangements in which Ali and Bilal do not sit next to each other.[3]
(b)Ali, Bilal and Chen must sit together, with Ali in the middle of the three. Find the number of arrangements.[2]
Show answer
(a) 30 240
total 8! = 40 320 B1; together 2 × 7! = 10 080 M1; 30 240 A1
(b) 1440
2 × 6! M1; 1440 A1
Q15Permutations · Challenge · Paper 1 style · no calculator · 4 marks
(a)Solve the equation ⁿP₃ = 20n, where n ≥ 3 is an integer.[4]
Q21Combinations · Standard · Paper 2 style · calculator · 2 marks
(a)A cricket squad of 11 is chosen from 15 players. The captain must be included and one injured player cannot be chosen. Find the number of possible squads.[2]
Show answer
(a) 286
choose 10 from 13 M1; ¹³C₁₀ = 286 A1
Q22Combinations · Standard · Paper 2 style · calculator · 4 marks
(a)A student chooses 4 books from a shelf holding 5 different physics books, 4 different chemistry books and 3 different biology books. Find the number of selections that contain at least one book of each subject.[4]
Q24Combinations · Standard · Paper 2 style · calculator · 4 marks
(a)In an examination, a candidate must answer 6 of the 9 questions. Questions 1 and 2 are compulsory. Find the number of ways of choosing the 6 questions.[2]
(b)Instead, there are no compulsory questions, but the candidate must answer exactly 4 of the first 5 questions. Find the number of ways.[2]
Show answer
(a) 35
choose 4 from 7 M1; 35 A1
(b) 30
⁵C₄ × ⁴C₂ M1; 30 A1
Q25Combinations · Standard · Paper 2 style · calculator · 4 marks
(a)12 points are marked on a page so that no three of them lie on a straight line. Find the number of straight lines that can be drawn through pairs of these points.[1]
(b)Find the number of triangles whose vertices are three of the points.[1]
(c)Find the number of these triangles that have a particular point P as one vertex.[2]
Show answer
(a) 66
¹²C₂ = 66 B1
(b) 220
¹²C₃ = 220 B1
(c) 55
¹¹C₂ M1; 55 A1
Q26Combinations · Standard · Paper 1 style · no calculator · 3 marks
(a)A medical team of 4 is chosen from 5 doctors and 4 nurses. Find the number of teams that include at least one doctor and at least one nurse.[3]
Show answer
(a) 120
total ⁹C₄ = 126 B1; all doctors 5, all nurses 1 M1; 126 − 6 = 120 A1
Q27Combinations · Challenge · Paper 2 style · calculator · 7 marks
(a)A committee of 6 is chosen from 7 men and 5 women. Find the number of committees with at least 2 women.[3]
(b)Mr and Mrs Khan are among the 12 people and refuse to serve together. Find the number of committees with at least 2 women that do not contain both of them.[4]
Show answer
(a) 812
total ¹²C₆ = 924 M1; minus no women ⁷C₆ = 7 and one woman 5 × ⁷C₅ = 105 M1; 812 A1
(b) 617
committees containing both, with at least one more woman: choose 4 from the other 6 men and 4 women with ≥ 1 woman: ¹⁰C₄ − ⁶C₄ = 195 M1 A1; 812 − 195 M1; 617 A1
Q28Combinations · Challenge · Paper 1 style · no calculator · 4 marks
(a)Show that ⁿC₂ + ⁽ⁿ⁺¹⁾C₂ = n² for n ≥ 2.[3]
(b)Hence find n such that ⁿC₂ + ⁽ⁿ⁺¹⁾C₂ = 169.[1]
Show answer
(a) n(n − 1)/2 + (n + 1)n/2 = n(2n)/2 = n²
both expressions in terms of n M1 M1; correct simplification A1
(b) n = 13
n² = 169, n = 13 B1
Q29Combinations · Challenge · Paper 2 style · calculator · 8 marks
(a)A delegation of 5 students is chosen from 4 students from Lahore, 5 from Karachi and 3 from Islamabad. Find the number of delegations with no restrictions.[1]
(b)Find the number of delegations that contain at least one student from each city.[3]
(c)Find the number of delegations that contain more students from Karachi than from Lahore.[4]
Show answer
(a) 792
¹²C₅ = 792 B1
(b) 590
complement: missing a city: no Lahore ⁸C₅ = 56, no Karachi ⁷C₅ = 21, no Islamabad ⁹C₅ = 126 M1; the delegation of 5 Karachi students (no Lahore and no Islamabad) is counted twice, so subtract 56 + 21 + 126 − 1 = 202 A1; 792 − 202 = 590 A1
(c) 376
cases (Lahore, Karachi, Islamabad): L = 0: K = 2..5 gives 10×1 + 10×3 + 5×3 + 1 = 56; L = 1: K = 2..4 gives 4(10×3 + 10×3 + 5) = 260; L = 2: K = 3 gives 6×10 = 60 M1 A1 A1; total 376 A1
Q30Combinations · Challenge · Paper 2 style · calculator · 5 marks
(a)4 boys are chosen from 7 boys and 3 girls are chosen from 5 girls. The 7 chosen children then stand in a row so that no two girls are next to each other. Find the number of ways this can be done.[5]
