Probability of a specific student being selected for a specific role when multiple...

1. Identify the total number of students: There are 24 students in the class. This will be the denominator for our probability calculation.

1. Identify the number of positions to be filled: The teacher selects 4 students for 4 distinct positions: president, vice president, secretary, and treasurer. Since the positions are distinct, the order matters, so we will use permutations.

1. Calculate the total number of ways to select and assign the 4 positions: The total number of ways to choose and assign 4 students from 24 to these specific roles is given by the permutation formula $P(n, k) = \frac{n!}{(n-k)!}$, where $n$ is the total number of items to choose from, and $k$ is the number of items to choose. In this case, $n=24$ and $k=4$. So, the total number of possible outcomes is $P(24, 4) = \frac{24!}{(24-4)!} = \frac{24!}{20!} = 24 \times 23 \times 22 \times 21$.

1. Determine the number of ways Nia can be chosen as president: For Nia to be chosen as president, she must occupy the president's role. There is only 1 way for Nia to be president. The remaining 3 positions (vice president, secretary, treasurer) need to be filled by the remaining 23 students. The number of ways to choose and assign these 3 positions from the remaining 23 students is $P(23, 3) = \frac{23!}{(23-3)!} = \frac{23!}{20!} = 23 \times 22 \times 21$.

1. Calculate the probability: The probability of an event is the number of favorable outcomes divided by the total number of possible outcomes. In this case, the favorable outcome is Nia being president. The number of ways Nia can be president is $1 \times P(23, 3)$. The total number of ways to fill the positions is $P(24, 4)$.

Therefore, the probability that Nia is chosen as president is:
$$P(\text{Nia is president}) = \frac{\text{Number of ways Nia is president}}{\text{Total number of ways to fill positions}}$$
$$P(\text{Nia is president}) = \frac{1 \times P(23, 3)}{P(24, 4)}$$
$$P(\text{Nia is president}) = \frac{23 \times 22 \times 21}{24 \times 23 \times 22 \times 21}$$
We can cancel out the common terms $23 \times 22 \times 21$ from the numerator and the denominator:
$$P(\text{Nia is president}) = \frac{1}{24}$$