Probability of a specific student being chosen for a specific role when multiple ro...

1. Identify the total number of possible outcomes: The teacher selects 4 students for 4 distinct positions (president, vice president, secretary, treasurer) from a class of 24 students. Since the positions are distinct, the order matters. This is a permutation problem. The total number of ways to select and arrange 4 students from 24 is given by the permutation formula $P(n, k) = \frac{n!}{(n-k)!}$. In this case, $n=24$ and $k=4$. So, the total number of possible outcomes is $P(24, 4) = \frac{24!}{(24-4)!} = \frac{24!}{20!} = 24 \times 23 \times 22 \times 21$. This is also represented as $_{24}P_4$.

1. Identify the number of favorable outcomes: We want to find the probability that Nia is chosen as president. If Nia is the president, there is only 1 way for this to happen for the president position. The remaining 3 positions (vice president, secretary, treasurer) must be filled by the other 23 students. The number of ways to select and arrange 3 students from the remaining 23 for the remaining 3 positions is $P(23, 3) = \frac{23!}{(23-3)!} = \frac{23!}{20!} = 23 \times 22 \times 21$. This is also represented as $_{23}P_3$.

3. Calculate the probability: The probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes.
Probability (Nia is president) = (Number of ways Nia is president and other positions are filled) / (Total number of ways to fill all positions)
Probability = $\frac{P(23, 3)}{P(24, 4)}$
Probability = $\frac{23 \times 22 \times 21}{24 \times 23 \times 22 \times 21}$

4. Simplify the expression: We can cancel out the common terms in the numerator and the denominator.
Probability = $\frac{1}{24}$

Alternatively, we can think of this problem by considering only the position of president. There are 24 students, and any one of them can be chosen as president. The probability that Nia is chosen as president is simply 1 out of 24, or $\frac{1}{24}$. The selection of the other positions does not affect the probability of Nia being chosen as president, as long as the selection process is random and independent for each position.