Examkin

Soru

Zorluk: ZorPermutations and Linear Arrangements

A panel of 6 distinct experts—3 scientists, 2 economists, and 1 moderator—are to sit in a single row of 6 chairs for a discussion. If the 3 scientists must all sit in adjacent chairs and the 2 economists cannot sit in adjacent chairs, how many different seating arrangements are possible?

  1. A
    12
  2. B
    36
  3. 72Cevap
  4. D
    108
  5. E
    144

Cevap

72 seating arrangements
To find the number of valid seating arrangements, we first group the 3 scientists together as 1 unit, leaving us with 2 non-economist entities (the scientist block and the moderator). These 2 entities can be arranged in 2! = 2 ways. Placing the 2 distinct economists into the 3 available gaps around these entities ensures they are not adjacent, yielding P(3, 2) = 6 choices. Finally, multiplying by the 3! = 6 internal arrangements of the scientists yields a total of 2 * 6 * 6 = 72 valid arrangements.

Adım Adım Çözüm

1
Group the 3 scientists into a single block SS, and treat the moderator MM as an individual unit.
There are 2 non-economist units: block SS and moderator MM.
Grouping elements that must be adjacent allows us to treat them temporarily as a single entity.
2
Calculate the arrangements of the non-economist units.
The 2 non-economist units can be arranged in 2!=22! = 2 ways.
Linear arrangement of 2 distinct entities.
3
Insert the 2 economists into the available gaps created by the non-economist units.
For any arrangement of SS and MM (e.g., _ SS _ MM _), there are 3 available gaps. The 2 distinct economists can be placed in these gaps in P(3,2)=3×2=6P(3,2) = 3 \times 2 = 6 ways.
To ensure no two economists sit together, each economist must occupy a separate gap.
4
Account for the internal arrangements of the 3 scientists within block SS.
The 3 distinct scientists can be arranged among themselves in 3!=63! = 6 ways.
Order matters among distinct individuals within a grouped block.
5
Apply the fundamental counting principle to compute total arrangements.
2×6×6=722 \times 6 \times 6 = 72 total arrangements.
Multiply the independent choices made in steps 2, 3, and 4.

Anahtar Kavram

Permutations with Adjacency and Non-Adjacency Restrictions
Bu soruyu puanla

Konular

Permutations and Linear Arrangements

Bu konudan daha fazla soru