In a social club with 1000 members, the executive committee would like to survey the members' opinion on the activities organized by the club. Give a possible reason why the committee would prefer to obtain a sample of opinions instead of asking all the members. Explain how a systematic sample of size 100 might be obtained. Give one disadvantage of using systematic sampling in this context. Ten friends at an event organized by a social club sit in a particular group of ten adjacent seats. Determine the number of possible seating arrangements in which two of them, Adam and Bernice, are separated by at least two people.

1. State a valid practical reason for choosing a sample over a census.

A sample is preferred because surveying the entire population of $1000$ members (a census) requires significantly more time, money, and administrative effort.

1. Determine the sampling interval $k$.

Population size, $N = 1000$
Sample size, $n = 100$
Sampling interval, $k = \frac{1000}{100} = 10$

2. Describe the full systematic sampling procedure.

1. Compile a numbered list of all $1000$ members.
2. Randomly select a starting number from $1$ to $10$ (for example, using a random number generator).
3. Select every $10$th member on the list thereafter until exactly $100$ members are chosen.

1. State a valid disadvantage of systematic sampling in the context of a membership list.

A key disadvantage of systematic sampling is its vulnerability to periodic or cyclical patterns in the sampling frame. If the club's membership list is structured such that similar types of members appear at regular intervals (for example, ordered by family units where every 10th person happens to be the head of a household), the resulting sample will not be representative of the entire club. Alternatively, one could note that unlike a simple random sample, not every possible group of 100 members has an equal chance of being selected (e.g., two members adjacent on the list can never be chosen together).

1. Calculate the total number of unrestricted seating arrangements.

The total number of ways the 10 friends can sit in 10 seats without any restrictions is:
$10! = 3,628,800$

2. Calculate the number of arrangements where Adam and Bernice sit together (separated by 0 people).

Treat Adam and Bernice as a single, inseparable unit. There are now $9$ units to arrange (this 1 unit + the $8$ other friends).
The 9 units can be arranged in $9!$ ways, and Adam and Bernice can swap positions within their unit in $2!$ ways.
$\text{Number of arrangements} = 9! \times 2! = 725,760$

3. Calculate the number of arrangements where Adam and Bernice are separated by exactly 1 person.

Choose $1$ out of the $8$ other friends to sit exactly between Adam and Bernice ($8$ possible choices).
Treat this block of 3 people as a single unit. Within this block, Adam and Bernice can swap places in $2!$ ways (the chosen friend remains in the middle).
There are now $8$ units to arrange in total (this new 3-person unit + the remaining $7$ friends).
$\text{Number of arrangements} = 8 \times 2! \times 8! = 645,120$

4. Subtract the complementary scenarios from the total to find the valid arrangements.

To find the number of ways they are separated by at least 2 people, subtract the number of ways they are separated by $0$ or $1$ person from the total unrestricted ways:
$\text{Valid arrangements} = 3,628,800 - 725,760 - 645,120$
$= 2,257,920$