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Zorluk: OrtaPermutations and Combinations

A panel discussion features 6 participants: 3 scientists, 2 economists, and 1 moderator. In how many different linear seating arrangements can these 6 participants be seated in a single row of 6 chairs if all 3 scientists must sit in adjacent seats?

  1. A
    2424
  2. B
    3636
  3. C
    120120
  4. 144144Cevap
  5. E
    720720

Cevap

The total number of valid seating arrangements is 144.
To arrange participants under the condition that all 3 scientists sit together, treat the 3 scientists as a single block. This leaves 4 units to arrange: 1 block of scientists, 2 economists, and 1 moderator. These 4 units can be arranged in a row in 4!=244! = 24 ways. Furthermore, within the scientist block, the 3 individual scientists can be arranged in 3!=63! = 6 distinct orders. By the fundamental counting principle, multiplying the arrangements of the main units by the internal arrangements of the block gives 24×6=14424 \times 6 = 144 total valid seating arrangements.

Adım Adım Çözüm

1
Group the restricted items into a single block.
Treat the 3 scientists as 1 single block unit.
Because the 3 scientists must sit together in adjacent seats, they move as a single block along with the other participants.
2
Calculate the number of arrangements for the distinct units.
There are 4 units to arrange (1 scientist block + 2 economists + 1 moderator), giving 4!=244! = 24 arrangements.
The Fundamental Counting Principle states that nn distinct items can be linearly arranged in n!n! ways.
3
Calculate the internal arrangements within the block.
The 3 scientists can be arranged among themselves within their block in 3!=63! = 6 ways.
The order of the individual scientists inside the block matters.
4
Multiply the unit arrangements by the internal arrangements.
Total arrangements = 4!×3!=24×6=1444! \times 3! = 24 \times 6 = 144.
By the multiplication principle, the total ways to complete both independent arrangement tasks is the product of their individual possibilities.

Anahtar Kavram

Permutations with Adjacency Restrictions (Block Method)
Tahmini Süre:1m 30s
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Konular

Permutations and Combinations

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