Probability and Combinatorics Quiz

Grade 10 · Math · Quiz

Challenging questions on counting methods, permutations, combinations, and probability rules.

Questions

1. In how many ways can 3 books be arranged on a shelf from a collection of 8 distinct books?

   1. 336
   2. 56
   3. 512
   4. 24
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   Answer: 336

   P(8,3) = 8!/(8−3)! = 8×7×6 = 336 ordered arrangements.

2. A die is rolled twice. What is the probability of getting a sum of 7?

   1. 1/6
   2. 1/12
   3. 7/36
   4. 1/3
   Show answer

   Answer: 1/6

   Pairs summing to 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) — 6 outcomes out of 36 total: 6/36 = 1/6.

3. A standard deck of 52 cards. What is the probability of drawing an ace or a red card?

   1. 7/13
   2. 4/13
   3. 1/2
   4. 3/13
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   Answer: 7/13

   P(ace ∪ red) = P(ace) + P(red) − P(red ace) = 4/52 + 26/52 − 2/52 = 28/52 = 7/13.

4. How many ways can a president, vice-president, and secretary be chosen from 12 candidates?

   1. 1320
   2. 220
   3. 1728
   4. 132
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   Answer: 1320

   P(12,3) = 12×11×10 = 1320 (ordered positions, so permutation).

5. C(n, r) = C(n, n − r) for all valid values of n and r.

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   Answer: True

   Choosing r items is equivalent to excluding (n−r) items, so C(n,r) = C(n,n−r) by symmetry of the formula.

6. Events A and B are independent. P(A) = 0.4 and P(B) = 0.3. What is P(A ∩ B)?

   1. 0.12
   2. 0.7
   3. 0.58
   4. 0.1
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   Answer: 0.12

   For independent events: P(A ∩ B) = P(A) × P(B) = 0.4 × 0.3 = 0.12.

7. A committee of 5 is chosen from 6 men and 4 women. How many committees contain exactly 3 men?

   1. 120
   2. 60
   3. 240
   4. 80
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   Answer: 120

   Choose 3 men from 6: C(6,3)=20. Choose 2 women from 4: C(4,2)=6. Total = 20×6 = 120.

8. The value of C(7, 3) is ______.

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   Answer: 35

   C(7,3) = 7!/(3!×4!) = (7×6×5)/(3×2×1) = 210/6 = 35.

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