$Number of Arrangements for a School Photo with Constraints on Teacher Placement$

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$Number of Arrangements for a School Photo with Constraints on Teacher Placement$

1. A schoolhas 11 teachers: 6 Math/Science teachers, 4 Humanities teachers, and 1 Phys-ed
teacher. They are arranging themselves in a row for a year-end photo.

a. Determine how many ways the teachers can be arranged for the photo if the Phys-ed
teacher stands in the middle, and two Math/Science teachers cannot stand next to each
other. Show your work.

b. Determine how many ways the teachers can be arranged for the photo if the
Humanities teachers want to all stand together. Show your work.

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John Bruce

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Savionf

+3

Very low bounty! Answering this question would take about 20 minutes. An offer of $2 means that you are expecting an expert to work for half the minimum wage!
- Naturelover
  
  +4
  
  It’s even less than that as the website gets 20% of it! And minimum wage is higher in other states. Unfortunately, this user wants to do all his questions for very cheap.
Erdos

+1

You should offer at least $10 for this kind of questions.

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The answer is accepted.

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Savionf

+3

Very low bounty! Answering this question would take about 20 minutes. An offer of $2 means that you are expecting an expert to work for half the minimum wage!

Naturelover

+4

It’s even less than that as the website gets 20% of it! And minimum wage is higher in other states. Unfortunately, this user wants to do all his questions for very cheap.
Naturelover

+4

It’s even less than that as the website gets 20% of it! And minimum wage is higher in other states. Unfortunately, this user wants to do all his questions for very cheap.
Erdos

+1

You should offer at least $10 for this kind of questions.

Answer 1

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a. The Phys-ed teacher should stand in the middle. 5 teachers will be standing in the left-hand side of the Phys-ed teacher, and 5 teachers in the right hand side. Since we want no two Math/Science teachers to stand next to each other. Math/Science teachers should be standing in positions $1, 3, 5, 7, 9, 11$. There are $6!$ ways that Math/Science teachers can stand in these 6 positions. The 4 Humanities teachers can stand in the remaining positions $(2,4,8, 10)$ in $4!$ ways. So the total number of ways is
\[6! \cdot 4!\]

b. If the Humanities teachers want to all stand together, we can think of them as a block of 4 teachers. So we have the remaining 7 teachers, and 1 block of Humanities teachers, will form 8 blocks which can be arranges in $8!$ ways. The 4 Humanities teachers can stand in the block in $4!$ ways. So the total number of ways is
\[4! \cdot 8!\]

Erdos

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Erdos

0

I was hesitant to accept the question, but since you are a new member I decide to answer :) Please offer higher bounties for your future posts, otherwise your questions make not get answered.
- John Bruce
  
  +1
  
  appreciate you g

$Number of Arrangements for a School Photo with Constraints on Teacher Placement$

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