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Zorluk: OrtaPermutations, Combinations, and Fundamental Counting Principle

An event coordinator is scheduling 5 distinct guest lectures—3 on Science and 2 on Art—to take place sequentially in 5 consecutive time slots. The coordinator establishes a restriction that the 2 Art lectures cannot be scheduled in consecutive time slots. Which of the following statements regarding the possible schedules are true? Select all such statements.

  1. The total number of possible arrangements for all 5 lectures without any restrictions is 120.Cevap
  2. B
    The number of schedules in which the 2 Art lectures are placed in consecutive time slots is 24.
  3. The total number of valid schedules in which the 2 Art lectures are not consecutive is 72.Cevap
  4. D
    The total number of valid schedules in which the 2 Art lectures are not consecutive is 36.
  5. E
    The total number of valid schedules in which the 2 Art lectures are not consecutive is 96.

Cevap

The statement specifying that the total unrestricted arrangements equal 120, and the statement specifying that the total valid non-consecutive schedules equal 72 are both correct.
Without restrictions, 5 distinct items can be linearly ordered in 5!=1205! = 120 ways. To find the number of ways where the 2 Art lectures are not adjacent, we subtract the ways they ARE adjacent (2!×4!=482! \times 4! = 48) from the total 120120, giving 120−48=72120 - 48 = 72. Alternatively, placing 3 Science lectures creates 4 available slots; selecting 2 slots and permuting the Art lectures yields 3!×(42)×2!=723! \times \binom{4}{2} \times 2! = 72.

Adım Adım Çözüm

1
Calculate total unrestricted arrangements of the 5 distinct lectures.
5!=1205! = 120 total arrangements.
The Fundamental Counting Principle specifies that 5 distinct items can be arranged in 5×4×3×2×1=1205 \times 4 \times 3 \times 2 \times 1 = 120 ways.
2
Calculate the number of restricted arrangements where the 2 Art lectures are placed consecutively.
2!×4!=482! \times 4! = 48 consecutive arrangements.
Treat the 2 Art lectures as a single combined block. The 3 Science lectures and 1 Art block form 4 units, which can be arranged in 4!=244! = 24 ways. Within the block, the 2 distinct Art lectures can be ordered in 2!=22! = 2 ways, yielding 24×2=4824 \times 2 = 48.
3
Subtract the restricted consecutive arrangements from the total arrangements to find the valid non-consecutive schedules.
120−48=72120 - 48 = 72 valid arrangements.
The complement rule allows finding non-consecutive placements by taking total arrangements minus consecutive arrangements.
4
Verify using the slot method (alternative approach).
3!×(42)×2!=6×6×2=723! \times \binom{4}{2} \times 2! = 6 \times 6 \times 2 = 72 valid arrangements.
Arrange 3 Science lectures in 3!=63! = 6 ways. This creates 4 potential slots between and around them (_ S1 _ S2 _ S3 _). Choosing 2 slots for the Art lectures takes (42)=6\binom{4}{2} = 6 ways, and arranging the 2 distinct Art lectures in those slots takes 2!=22! = 2 ways.

Anahtar Kavram

Permutations with Adjacency Restrictions and Complementary Counting
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Permutations, Combinations, and Fundamental Counting Principle

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