COMC COMC Geometry and Combinatorics 2 ā Questions and Answers
Question 1: In how many ways can 5 distinct books be arranged on a shelf?
- 120 (Correct answer)
- 60
- 24
- 100
Correct answer: 120
The number of permutations of 5 distinct objects is 5! = 5 Ć 4 Ć 3 Ć 2 Ć 1 = 120.
Question 2: How many ways can a committee of 3 people be chosen from a group of 12?
- 220 (Correct answer)
- 132
- 360
- 66
Correct answer: 220
C(12,3) = 12!/(3!Ć9!) = (12Ć11Ć10)/(3Ć2Ć1) = 220.
Question 3: How many 4-digit numbers can be formed using digits 1ā9 without repetition?
- 3024 (Correct answer)
- 6561
- 504
- 9000
Correct answer: 3024
This is P(9,4) = 9 Ć 8 Ć 7 Ć 6 = 3024 ordered arrangements.
Question 4: If two fair six-sided dice are rolled, in how many ways can the sum equal 7?
- 6 (Correct answer)
- 5
- 7
- 8
Correct answer: 6
The pairs are (1,6),(2,5),(3,4),(4,3),(5,2),(6,1), giving exactly 6 ordered outcomes.
Question 5: In how many ways can 8 people be split into two unlabeled groups of 4?
- 35 (Correct answer)
- 70
- 56
- 28
Correct answer: 35
Choose 4 people in C(8,4) = 70 ways, then divide by 2 because swapping the two groups gives the same split: 70/2 = 35.
Question 6: How many distinct arrangements are there of the letters in the word LEVEL?
- 30 (Correct answer)
- 60
- 120
- 20
Correct answer: 30
LEVEL has 5 letters with L appearing twice and E appearing twice, giving 5!/(2!Ć2!) = 120/4 = 30.
COMC COMC Geometry and Combinatorics 2
In how many ways can 5 distinct books be arranged on a shelf?