Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
In a class of 10 students if two prizes (1st and 2nd) has to be given in three subjects. Physics, Chemistry & Mathematics and this can be done in $k$ ways, then $\frac{k}{1000}$ is_____.
Step-by-Step Solution
Key Concept: Each subject independently requires selecting an ordered pair of students for 1st and 2nd prizes, giving $P(10,2)$ ways per subject, then multiply across all 3 subjects.
For each subject, we need to select 2 students from 10 for 1st and 2nd prizes, where order matters. For one subject, this can be done in $P(10,2) = 10 \times 9 = 90$ ways. Since there are 3 independent subjects (Physics, Chemistry, Mathematics), the total number of ways is $k = 90 \times 90 \times 90 = 90^3 = 729000$. Therefore, $\frac{k}{1000} = \frac{729000}{1000} = 729$.
Correct Answer: 729