Question

the track team gives awards for first, second, and third place runners. there are 10 students from school a and 12 students from school b competing. which expression represents the probability that all three awards will go to a student from school b? \\(\frac{_{12}p_3}{_{22}p_3}\\) \\(\frac{_{12}c_3}{_{22}c_3}\\) \\(\frac{_{22}p_3}{_{22}p_{12}}\\) \\(\frac{_{22}c_3}{_{22}c_{12}}\\)

Explanation:

Step1: Understand Permutations vs Combinations

Since the awards are for first, second, and third place, the order matters (first, second, third are distinct positions). So we use permutations (\( _nP_r \)) instead of combinations (\( _nC_r \)).

Step2: Determine Total and Favorable Outcomes

• Total number of ways to award 3 distinct places (first, second, third) among all \( 10 + 12 = 22 \) students: \( _{22}P_3 \).
• Favorable number of ways: award all 3 places to school B students (12 students), so \( _{12}P_3 \).

Step3: Probability Formula

Probability is favorable outcomes divided by total outcomes, so \( \frac{_{12}P_3}{_{22}P_3} \).

Answer:

\(\boldsymbol{\frac{_{12}P_3}{_{22}P_3}}\) (the first option)