AMC 8 Challenge Lab
Lesson 9: Counting Principles & Combinatorics
Recognize • Model • Solve • Verify
50 original AMC 8-style multiple-choice problems with complete English solutions.
AMC 8 Strategy
Build speed without sacrificing accuracy.
1. Read for the exact target and hidden restrictions.
2. Translate words into a diagram, equation, table, or smaller case.
3. Estimate before calculating; eliminate impossible choices.
4. Look for shortcuts: symmetry, factoring, parity, ratios, or patterns.
5. Verify the result and make sure it answers the question asked.
Lesson Focus
Counting Principles & Combinatorics
• Concept recognition
• Efficient contest strategy
• Mental-math checks
• Multiple representations
• Answer-choice verification
Practice 1-2
Choose the best answer. Solutions begin after Question 50.
1. There are 6 shirt choices, 7 pants choices and 2 hat choices. How many outfits are possible?��A. 86 B. 42 C. 84 D. 15 E. 82
2. A bag has 6 red and 5 blue marbles. What is the probability of choosing a red marble?��A. 6/12 B. 5/11 C. 1/11 D. 6/11 E. 6/5
Practice 3-4
Choose the best answer. Solutions begin after Question 50.
3. The arithmetic sequence starts 2, 9, 16, ... . What is the 9th term?��A. 51 B. 63 C. 65 D. 58 E. 63
4. What is the greatest common factor of 36 and 32?��A. 288 B. 4 C. 10 D. 8 E. 2
Practice 5-6
Choose the best answer. Solutions begin after Question 50.
5. What is 25×21−25?��A. 505 B. 525 C. 500 D. 550 E. 479
6. There are 6 shirt choices, 4 pants choices and 4 hat choices. How many outfits are possible?��A. 100 B. 14 C. 92 D. 24 E. 96
Practice 7-8
Choose the best answer. Solutions begin after Question 50.
7. A bag has 5 red and 8 blue marbles. What is the probability of choosing a red marble?��A. 8/13 B. 5/14 C. 5/13 D. 5/8 E. 1/13
8. The arithmetic sequence starts 2, 8, 14, ... . What is the 14th term?��A. 74 B. 86 C. 80 D. 84 E. 85
Practice 9-10
Choose the best answer. Solutions begin after Question 50.
9. What is the greatest common factor of 24 and 32?��A. 4 B. 16 C. 14 D. 8 E. 96
10. What is 12×16−12?��A. 204 B. 192 C. 180 D. 185 E. 164
Practice 11-12
Choose the best answer. Solutions begin after Question 50.
11. There are 6 shirt choices, 5 pants choices and 3 hat choices. How many outfits are possible?��A. 93 B. 30 C. 90 D. 14 E. 87
12. A bag has 4 red and 4 blue marbles. What is the probability of choosing a red marble?��A. 4/4 B. 1/8 C. 4/8 D. 4/9 E. 4/8
Practice 13-14
Choose the best answer. Solutions begin after Question 50.
13. The arithmetic sequence starts 2, 7, 12, ... . What is the 11th term?��A. 52 B. 47 C. 55 D. 57 E. 57
14. What is the greatest common factor of 54 and 32?��A. 2 B. 1 C. 8 D. 4 E. 864
Practice 15-16
Choose the best answer. Solutions begin after Question 50.
15. What is 17×11−17?��A. 204 B. 187 C. 175 D. 159 E. 170
16. There are 6 shirt choices, 6 pants choices and 2 hat choices. How many outfits are possible?��A. 36 B. 14 C. 70 D. 72 E. 74
Practice 17-18
Choose the best answer. Solutions begin after Question 50.
17. A bag has 3 red and 7 blue marbles. What is the probability of choosing a red marble?��A. 3/10 B. 1/10 C. 3/11 D. 7/10 E. 3/7
18. The arithmetic sequence starts 2, 6, 10, ... . What is the 8th term?��A. 30 B. 35 C. 32 D. 34 E. 26
Practice 19-20
Choose the best answer. Solutions begin after Question 50.
19. What is the greatest common factor of 42 and 32?��A. 672 B. 2 C. 1 D. 8 E. 4
20. What is 22×21−22?��A. 484 B. 419 C. 440 D. 462 E. 445
Practice 21-22
Choose the best answer. Solutions begin after Question 50.
21. There are 6 shirt choices, 7 pants choices and 4 hat choices. How many outfits are possible?��A. 168 B. 164 C. 17 D. 42 E. 172
22. A bag has 2 red and 3 blue marbles. What is the probability of choosing a red marble?��A. 1/5 B. 2/3 C. 2/5 D. 3/5 E. 2/6
Practice 23-24
Choose the best answer. Solutions begin after Question 50.
23. The arithmetic sequence starts 2, 5, 8, ... . What is the 13th term?��A. 43 B. 41 C. 35 D. 38 E. 39
24. What is the greatest common factor of 30 and 32?��A. 480 B. 4 C. 2 D. 8 E. 1
Practice 25-26
Choose the best answer. Solutions begin after Question 50.
25. What is 27×16−27?��A. 405 B. 389 C. 410 D. 432 E. 459
26. There are 6 shirt choices, 4 pants choices and 3 hat choices. How many outfits are possible?��A. 13 B. 72 C. 75 D. 69 E. 24
Practice 27-28
Choose the best answer. Solutions begin after Question 50.
27. A bag has 7 red and 6 blue marbles. What is the probability of choosing a red marble?��A. 7/13 B. 6/13 C. 7/6 D. 7/14 E. 1/13
28. The arithmetic sequence starts 2, 4, 6, ... . What is the 10th term?��A. 20 B. 18 C. 25 D. 24 E. 22
Practice 29-30
Choose the best answer. Solutions begin after Question 50.
29. What is the greatest common factor of 18 and 32?��A. 4 B. 1 C. 288 D. 8 E. 2
30. What is 14×11−14?��A. 145 B. 168 C. 129 D. 154 E. 140
Practice 31-32
Choose the best answer. Solutions begin after Question 50.
31. There are 6 shirt choices, 5 pants choices and 2 hat choices. How many outfits are possible?��A. 13 B. 62 C. 58 D. 60 E. 30
32. A bag has 6 red and 9 blue marbles. What is the probability of choosing a red marble?��A. 6/9 B. 6/15 C. 1/15 D. 9/15 E. 6/16
Practice 33-34
Choose the best answer. Solutions begin after Question 50.
33. The arithmetic sequence starts 2, 9, 16, ... . What is the 15th term?��A. 105 B. 93 C. 107 D. 100 E. 105
34. What is the greatest common factor of 48 and 32?��A. 22 B. 16 C. 8 D. 96 E. 32
Practice 35-36
Choose the best answer. Solutions begin after Question 50.
35. What is 19×21−19?��A. 359 B. 418 C. 380 D. 385 E. 399
36. There are 6 shirt choices, 6 pants choices and 4 hat choices. How many outfits are possible?��A. 16 B. 36 C. 148 D. 144 E. 140
Practice 37-38
Choose the best answer. Solutions begin after Question 50.
37. A bag has 5 red and 5 blue marbles. What is the probability of choosing a red marble?��A. 5/11 B. 5/10 C. 1/10 D. 5/5 E. 5/10
38. The arithmetic sequence starts 2, 8, 14, ... . What is the 12th term?��A. 73 B. 62 C. 74 D. 72 E. 68
Practice 39-40
Choose the best answer. Solutions begin after Question 50.
39. What is the greatest common factor of 36 and 32?��A. 2 B. 8 C. 10 D. 4 E. 288
40. What is 24×16−24?��A. 408 B. 365 C. 384 D. 360 E. 344
Practice 41-42
Choose the best answer. Solutions begin after Question 50.
41. There are 6 shirt choices, 7 pants choices and 3 hat choices. How many outfits are possible?��A. 42 B. 123 C. 16 D. 129 E. 126
42. A bag has 4 red and 8 blue marbles. What is the probability of choosing a red marble?��A. 1/12 B. 4/8 C. 4/12 D. 4/13 E. 8/12
Practice 43-44
Choose the best answer. Solutions begin after Question 50.
43. The arithmetic sequence starts 2, 7, 12, ... . What is the 9th term?��A. 45 B. 42 C. 47 D. 47 E. 37
44. What is the greatest common factor of 24 and 32?��A. 16 B. 96 C. 8 D. 14 E. 4
Practice 45-46
Choose the best answer. Solutions begin after Question 50.
45. What is 29×11−29?��A. 348 B. 319 C. 279 D. 295 E. 290
46. There are 6 shirt choices, 4 pants choices and 2 hat choices. How many outfits are possible?��A. 50 B. 48 C. 46 D. 12 E. 24
Practice 47-48
Choose the best answer. Solutions begin after Question 50.
47. A bag has 3 red and 4 blue marbles. What is the probability of choosing a red marble?��A. 4/7 B. 3/4 C. 3/7 D. 1/7 E. 3/8
48. The arithmetic sequence starts 2, 6, 10, ... . What is the 14th term?��A. 59 B. 50 C. 54 D. 58 E. 56
Practice 49-50
Choose the best answer. Solutions begin after Question 50.
49. What is the greatest common factor of 54 and 32?��A. 864 B. 2 C. 8 D. 1 E. 4
50. What is 16×21−16?��A. 325 B. 352 C. 299 D. 336 E. 320
Answer Key
Questions 1-50
1. C�2. D�3. D�4. B�5. C�6. E�7. C�8. C�9. D�10. C
11. C�12. C�13. A�14. A�15. E�16. D�17. A�18. A�19. B�20. C
21. A�22. C�23. D�24. C�25. A�26. B�27. A�28. A�29. E�30. E
31. D�32. B�33. D�34. B�35. C�36. D�37. B�38. E�39. D�40. D
41. E�42. C�43. B�44. C�45. E�46. B�47. C�48. C�49. B�50. E
Solutions 1-2
Check the method, not just the letter.
1. Answer: C. 84��Multiplication principle: 6×7×2=84.
2. Answer: D. 6/11��There are 11 total marbles and 6 red, so the probability is 6/11.
Solutions 3-4
Check the method, not just the letter.
3. Answer: D. 58��a_n=a_1+(n−1)d=2+(9−1)(7)=58.
4. Answer: B. 4��Prime-factor or Euclidean-algorithm work gives gcd(36,32)=4.
Solutions 5-6
Check the method, not just the letter.
5. Answer: C. 500��Factor 25: 25(21−1)=25×20=500.
6. Answer: E. 96��Multiplication principle: 6×4×4=96.
Solutions 7-8
Check the method, not just the letter.
7. Answer: C. 5/13��There are 13 total marbles and 5 red, so the probability is 5/13.
8. Answer: C. 80��a_n=a_1+(n−1)d=2+(14−1)(6)=80.
Solutions 9-10
Check the method, not just the letter.
9. Answer: D. 8��Prime-factor or Euclidean-algorithm work gives gcd(24,32)=8.
10. Answer: C. 180��Factor 12: 12(16−1)=12×15=180.
Solutions 11-12
Check the method, not just the letter.
11. Answer: C. 90��Multiplication principle: 6×5×3=90.
12. Answer: C. 4/8��There are 8 total marbles and 4 red, so the probability is 4/8.
Solutions 13-14
Check the method, not just the letter.
13. Answer: A. 52��a_n=a_1+(n−1)d=2+(11−1)(5)=52.
14. Answer: A. 2��Prime-factor or Euclidean-algorithm work gives gcd(54,32)=2.
Solutions 15-16
Check the method, not just the letter.
15. Answer: E. 170��Factor 17: 17(11−1)=17×10=170.
16. Answer: D. 72��Multiplication principle: 6×6×2=72.
Solutions 17-18
Check the method, not just the letter.
17. Answer: A. 3/10��There are 10 total marbles and 3 red, so the probability is 3/10.
18. Answer: A. 30��a_n=a_1+(n−1)d=2+(8−1)(4)=30.
Solutions 19-20
Check the method, not just the letter.
19. Answer: B. 2��Prime-factor or Euclidean-algorithm work gives gcd(42,32)=2.
20. Answer: C. 440��Factor 22: 22(21−1)=22×20=440.
Solutions 21-22
Check the method, not just the letter.
21. Answer: A. 168��Multiplication principle: 6×7×4=168.
22. Answer: C. 2/5��There are 5 total marbles and 2 red, so the probability is 2/5.
Solutions 23-24
Check the method, not just the letter.
23. Answer: D. 38��a_n=a_1+(n−1)d=2+(13−1)(3)=38.
24. Answer: C. 2��Prime-factor or Euclidean-algorithm work gives gcd(30,32)=2.
Solutions 25-26
Check the method, not just the letter.
25. Answer: A. 405��Factor 27: 27(16−1)=27×15=405.
26. Answer: B. 72��Multiplication principle: 6×4×3=72.
Solutions 27-28
Check the method, not just the letter.
27. Answer: A. 7/13��There are 13 total marbles and 7 red, so the probability is 7/13.
28. Answer: A. 20��a_n=a_1+(n−1)d=2+(10−1)(2)=20.
Solutions 29-30
Check the method, not just the letter.
29. Answer: E. 2��Prime-factor or Euclidean-algorithm work gives gcd(18,32)=2.
30. Answer: E. 140��Factor 14: 14(11−1)=14×10=140.
Solutions 31-32
Check the method, not just the letter.
31. Answer: D. 60��Multiplication principle: 6×5×2=60.
32. Answer: B. 6/15��There are 15 total marbles and 6 red, so the probability is 6/15.
Solutions 33-34
Check the method, not just the letter.
33. Answer: D. 100��a_n=a_1+(n−1)d=2+(15−1)(7)=100.
34. Answer: B. 16��Prime-factor or Euclidean-algorithm work gives gcd(48,32)=16.
Solutions 35-36
Check the method, not just the letter.
35. Answer: C. 380��Factor 19: 19(21−1)=19×20=380.
36. Answer: D. 144��Multiplication principle: 6×6×4=144.
Solutions 37-38
Check the method, not just the letter.
37. Answer: B. 5/10��There are 10 total marbles and 5 red, so the probability is 5/10.
38. Answer: E. 68��a_n=a_1+(n−1)d=2+(12−1)(6)=68.
Solutions 39-40
Check the method, not just the letter.
39. Answer: D. 4��Prime-factor or Euclidean-algorithm work gives gcd(36,32)=4.
40. Answer: D. 360��Factor 24: 24(16−1)=24×15=360.
Solutions 41-42
Check the method, not just the letter.
41. Answer: E. 126��Multiplication principle: 6×7×3=126.
42. Answer: C. 4/12��There are 12 total marbles and 4 red, so the probability is 4/12.
Solutions 43-44
Check the method, not just the letter.
43. Answer: B. 42��a_n=a_1+(n−1)d=2+(9−1)(5)=42.
44. Answer: C. 8��Prime-factor or Euclidean-algorithm work gives gcd(24,32)=8.
Solutions 45-46
Check the method, not just the letter.
45. Answer: E. 290��Factor 29: 29(11−1)=29×10=290.
46. Answer: B. 48��Multiplication principle: 6×4×2=48.
Solutions 47-48
Check the method, not just the letter.
47. Answer: C. 3/7��There are 7 total marbles and 3 red, so the probability is 3/7.
48. Answer: C. 54��a_n=a_1+(n−1)d=2+(14−1)(4)=54.
Solutions 49-50
Check the method, not just the letter.
49. Answer: B. 2��Prime-factor or Euclidean-algorithm work gives gcd(54,32)=2.
50. Answer: E. 320��Factor 16: 16(21−1)=16×20=320.