Permutation and Combination Quiz

Challenge your understanding of permutation and combination with application-based questions designed to strengthen your grasp of fundamental counting principles, arrangements, and selections. This quiz covers key concepts, essential formulas, and real-world problem scenarios involving permutations and combinations.

1. Distinguishing Permutation from Combination

   Which of the following situations best illustrates a permutation: selecting top three winners from a group of ten participants?

   1. Yes, because the order of winners matters
   2. Yes, because the selection is random
   3. No, because only the group selected is important
   4. No, the order is irrelevant in this case

   Explanation: Selecting top three winners from ten participants is a permutation because the order in which the winners are picked (first, second, third place) matters. Combinations are used when only the grouping is important and the order does not matter, making option B and C incorrect. Option D misattributes randomness as the key factor, but it's actually the importance of order that distinguishes a permutation.

2. Applying the Permutation Formula

   How many different 3-letter passwords can be formed from the letters A, B, C, D if no letter is repeated and the order matters?

   1. 64
   2. 81
   3. 24
   4. 12

   Explanation: Since order matters and no repetition is allowed, use the permutation formula: 4 choices for the first letter, 3 for the second, and 2 for the third (4×3×2=24). Option B (12) only considers combinations without order, while 64 and 81 are results of other formulas, such as with repetition or combinations. Thus, only 24 is the correct count for ordered, non-repeating selections.

3. Choosing Committees (Combination Problem)

   From a club of 8 members, how many different 3-member committees can be formed?

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

   Explanation: When forming a committee where order doesn’t matter, the combination formula is used: 8 choose 3 (8C3) which equals 56. Option B (336) represents the permutation where order matters, option C (24) is incorrect for this scenario, and option D (512) does not relate to any relevant formula. Only 56 accurately reflects unordered selection.

4. Including Restrictions in Permutations

   How many ways can five students stand in a line if two specific students must stand together?

   1. 24
   2. 60
   3. 48
   4. 120

   Explanation: Treat the two specific students as a single entity, reducing the arrangement to 4 items. Arranging 4 items is 4! (24 ways), but the two together can switch places in 2! (2) ways, so 24×2=48. Option B (120) is all students arranged with no restriction, option C (24) ignores the switching, and option D (60) is unrelated. Only 48 meets the specifics of students standing together.

5. Permutations with Identical Objects

   In how many unique ways can the letters of the word 'LEVEL' be arranged?

   1. 20
   2. 30
   3. 60
   4. 120

   Explanation: 'LEVEL' has five letters with 'L' and 'E' each repeating twice. The correct calculation is 5! divided by (2! x 2!) for the identical letters, giving 120/(2×2)=30. Option B (60) ignores one set of repeats, C (120) treats all as unique, and D (20) miscalculates the division. Only 30 correctly accounts for indistinguishable letters.