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Zorluk: OrtaPermutations and Linear Arrangements

An event organizer is scheduling a conference featuring 77 distinct speakers: 33 technology experts, 33 finance experts, and 11 keynote presenter. The 77 speakers will give back-to-back presentations on a single main stage. If all 33 technology experts must speak consecutively in any order, and no two finance experts may speak consecutively, how many different presentation schedules are possible?

Cevap: 72 schedules

Cevap

The total number of possible presentation schedules satisfying all conditions is 72.
To find the number of valid schedules, we combine the block method and the gap method. First, bundle the 3 technology experts into a single block. This block can be ordered internally in 3!=63! = 6 ways. Next, arrange the non-finance items (the technology block and the keynote presenter), which can be ordered in 2!=22! = 2 ways. These 2 items create 3 available gaps (one at each end and one in the middle) for the 3 finance experts. To ensure no two finance experts are consecutive, place exactly one finance expert into each of the 3 gaps, which can be done in 3!=63! = 6 ways. Multiplying these independent choices yields 6×2×6=726 \times 2 \times 6 = 72 total valid schedules.

Adım Adım Çözüm

1
Group the 3 technology experts into a single unit and calculate internal permutations
3! = 6 internal arrangements for the technology block
The technology experts must appear consecutively, so treating them as a single block ensures they remain together.
2
Arrange the non-finance elements (the technology block and the keynote presenter)
2! = 2 linear arrangements
There are 2 distinct items that act as separators for the finance experts.
3
Apply the gap method to place the 3 finance experts into non-adjacent slots
3 available gaps for 3 finance experts gives 3! = 6 ways
Placing at most one finance expert in each gap created by the non-finance items guarantees that no two finance experts are seated or scheduled consecutively.
4
Calculate total permutations using the Fundamental Counting Principle
6 × 2 × 6 = 72 total schedules
The decisions for internal block order, base item order, and gap assignments are independent.

Anahtar Kavram

Permutations with Combined Block and Separation (Gap Method) Restrictions
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Konular

Permutations and Linear Arrangements

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