Question: An entomologist is studying a population of 12 bees, selecting a sample of 5 for genetic analysis. Among the bees, 4 are tagged for tracking, and we want to ensure at least one tagged bee is included in the sample. How many such samples are possible? - Treasure Valley Movers

How an Entomologist Ensures Tagged Bees Are Included When Sampling a Bee Population

Curiosity about pollinators and their vital role in ecosystems is growing across the United States, especially as scientists work to protect vulnerable bee populations. A common challenge in ecological studies involves selecting representative, genetically meaningful samples—particularly when tracking individual bees over time. In one such scenario, researchers tracking a colony of 12 bees must choose a sample of 5 for detailed genetic analysis. Among these 12, 4 bees carry unique tracking tags, and ensuring at least one tagged bee is included transforms this selection into a key statistical consideration. So, how many valid samples meet this requirement? This question reflects real-world priorities in biodiversity monitoring and provides a clear example of intentional sampling design.

---

The Science Behind Selective Sampling

When studying insect populations like bees, researchers rely on careful statistical sampling to draw accurate conclusions. Genetic studies, in particular, demand representative subsets that reflect the broader population’s diversity. Tagged bees often serve as identifiers in long-term monitoring, enabling scientists to track movement, health, and lineage. Random sampling without inclusion guarantees risks excluding these key individuals, potentially skewing data. Ensuring at least one tagged bee appears in a 5-bee sample means balancing random selection with intentional inclusion—making the math beneath the process both precise and meaningful.

---

How the Sample Is Calculated

To determine the number of ways to choose 5 bees from 12 with the condition that at least one of the 4 tagged bees is included, one straightforward approach uses complementary counting. Rather than listing all valid combinations directly, we first calculate all possible 5-bee samples, then subtract the scenarios where no tagged bees appear—an approach that simplifies complex inclusion logic.

Start with total samples:
Total ways to choose 5 bees from 12:
[ \binom{12}{5} = \frac{12!}{5!(12-5)!} = 792 ]

Now calculate invalid samples—those with zero tagged bees. Since there are 4 tagged bees, 8 are untagged. Choosing 5 bees with no tagged bees means selecting all 5 from the 8 untagged:
[ \binom{8}{5} = \frac{8!}{5!3!} = 56 ]

Subtract invalid from total:
[