An office team of 10 people includes 7 men and 3 women named Anne, Beth, and Cathie. For an upcoming fire drill exercise, 5 individuals will be chosen, each assigned a unique role, to carry out the drill. Determine the number of possible ways to select 5 people from this group of 10

1. Calculate the number of ways to choose 5 people out of 10 and assign them to unique roles

\text{Number of ways} = {}^{10}P_5 = 30,240

1. Calculate the number of ways to choose and assign roles for a team of 0 women (i.e. 5 men)

The condition 'at most 1 woman' splits into two mutually exclusive cases: exactly 0 women and exactly 1 woman.

For 0 women, we select 5 men from the 7 available and arrange them into 5 unique roles:
$^{7}\text{C}_{5} \times 5! = ^{7}\text{P}_{5} = 2520$

2. Calculate the number of ways to choose and assign roles for a team of exactly 1 woman and 4 men

For 1 woman, we select 1 woman from 3 and 4 men from 7, then arrange all 5 selected individuals into the 5 unique roles:
$^{3}\text{C}_{1} \times ^{7}\text{C}_{4} \times 5! = 3 \times 35 \times 120 = 12600$

3. Sum the cases to find the total number of valid ways

Total number of ways $= 2520 + 12600 = 15120$

1. Determine the number of internal arrangements for Anne, Beth, and Cathie.

Since Beth must be seated between Anne and Cathie, the valid arrangements for these three people are Anne-Beth-Cathie and Cathie-Beth-Anne.
Number of ways $= 2! = 2$

2. Calculate the total number of circular arrangements by treating the restricted group as a single unit.

Treat the group of three women as $1$ unit. There are $7$ remaining men, making a total of $1 + 7 = 8$ units to arrange around the round table.
The number of ways to arrange $n$ units in a circle is $(n-1)!$.
$\text{Total number of arrangements} = (8 - 1)! \times 2 = 7! \times 2 = 5040 \times 2 = 10080$

1. Find the total unrestricted number of arrangements for the 10 people.

$\text{Total ways} = 10! = 3,628,800$

2. Find the number of arrangements where Beth is standing beside Anne (and similarly for Cathie).

In each row of 5, there are 4 pairs of adjacent positions, giving $4 \times 2 = 8$ pairs in total.
Treating Beth and Anne as a block, they can occupy any of these 8 adjacent pairs and can be swapped internally in $2!$ ways.
The remaining 8 people are arranged in the remaining 8 seats in $8!$ ways.
$\text{Ways Beth is beside Anne} = 8 \times 2! \times 8! = 645,120$
Similarly, the number of ways Beth is beside Cathie is also $645,120$.

3. Find the number of arrangements where Beth is standing beside both Anne and Cathie.

For Beth to be beside both, they must form a 3-person block (Anne-Beth-Cathie or Cathie-Beth-Anne) in the same row.
In each row, there are 3 such blocks of three adjacent positions, giving $3 \times 2 = 6$ blocks in total.
Beth must strictly be in the middle, while Anne and Cathie can be swapped in $2!$ ways.
The remaining 7 people are arranged in the remaining 7 seats in $7!$ ways.
$\text{Ways Beth is beside both} = 6 \times 2! \times 7! = 60,480$

4. Use the principle of inclusion-exclusion and the complement method to find the required number of arrangements.

Using the principle of inclusion-exclusion, the number of ways Beth is beside either Anne or Cathie is:
$645,120 + 645,120 - 60,480 = 1,229,760$
Subtracting this from the unrestricted total gives the number of ways Beth is not beside either of them:
$\text{Required ways} = 3,628,800 - 1,229,760 = 2,399,040$