To calculate the probability of selecting 14 teachers with specific numbers from different age groups, we can use the multinomial probability formula. The formula is as follows:

$P = \frac{\binom{n_{30}}{2} \binom{n_{40}}{3} \binom{n_{50}}{9} \binom{n_{60}}{0}}{\binom{n_{total}}{14}}$

Where:

• $n_{30} = 10$ is the number of teachers in the 30-39 age group,
• $n_{40} = 20$ is the number of teachers in the 40-49 age group,
• $n_{50} = 30$ is the number of teachers in the 50-59 age group,
• $n_{60} = 15$ is the number of teachers in the 60-69 age group,
• $n_{total} = 10 + 20 + 30 + 15 = 75$ is the total number of teachers,
• We are selecting a total of 14 teachers: 2 from the 30s, 3 from the 40s, and 9 from the 50s.

Now, we will calculate this probability step by step.

Let me compute this.The probability of selecting 14 teachers such that 2 are in their 30s, 3 are in their 40s, and 9 are in their 50s is $0.0013$, or 0.13%.

Do you have any further questions or would you like more details on how this was calculated?

Here are 5 related questions you could explore:

1. How would the probability change if you selected 10 teachers instead of 14?
2. What is the probability of selecting 1 teacher from each age group in a group of 4?
3. How does the binomial coefficient help in probability calculations?
4. If the number of teachers aged 50-59 increased, how would that affect the probability?
5. What is the probability that all selected teachers are aged 40-49?

Tip: The multinomial coefficient helps when you're selecting items from multiple groups simultaneously, considering different proportions for each group.