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Zorluk: KolayPermutations and Linear Arrangements

Five speakers—Alice, Bob, Carol, David, and Eva—are scheduled to give consecutive presentations at a conference. If Alice must present first, in how many different linear orders can all five speakers present?

  1. A
    120
  2. B
    48
  3. 24Cevap
  4. D
    16
  5. E
    12

Cevap

The total number of different presentation orders is 24.
Since Alice's position is fixed as the first speaker, there is only 11 way to fill the first slot. The remaining 44 speakers can be arranged in the remaining 44 slots in 4!=4×3×2×1=244! = 4 \times 3 \times 2 \times 1 = 24 ways. Thus, the total number of linear arrangements is 1×24=241 \times 24 = 24.

Adım Adım Çözüm

1
Identify the fixed element and available positions.
Alice must fill the 1st position, leaving 1 choice for slot 1.
The question specifies that Alice must present first.
2
Calculate the number of ways to arrange the remaining speakers.
The remaining 4 speakers (Bob, Carol, David, Eva) can be arranged in 4!=4×3×2×1=244! = 4 \times 3 \times 2 \times 1 = 24 ways.
Linear permutations of nn distinct objects use n!n!.
3
Apply the Fundamental Counting Principle.
Total arrangements = 1×24=241 \times 24 = 24.
Combine the independent choices for each position.

Anahtar Kavram

Permutations with Fixed Positions
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Konular

Permutations and Linear Arrangements

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