Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
In an examination, a candidate is required to pass in all four subjects he is studying. The number of ways in which he can fail is:
$^4P_1 + ^4P_2 + ^4P_3 + ^4P_4$
$4^4 - 1$
$2^4 - 1$
$^4C_1 + ^4C_2 + ^4C_3 + ^4C_4$
Step-by-Step Solution
Key Concept: Failing means not passing in at least one subject, which is the complement of passing all four, giving $2^4 - 1$ total failure scenarios.
A candidate fails if he doesn't pass in at least one subject. The total number of ways to fail is the complement of passing all four subjects. Since each subject can either be passed or failed (2 outcomes), the total possibilities are $2^4$. Subtracting the one way to pass all subjects gives $2^4 - 1$ ways to fail. This also equals $^4C_1 + ^4C_2 + ^4C_3 + ^4C_4$ (fail in exactly 1, 2, 3, or 4 subjects respectively), which sums to $2^4 - 1$ by the binomial theorem.
Correct Answer: 3,4