Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
If the total number of strictly increasing functions defined from $f : \{1,2,3,4,5,6\} \to \{1,2,3,...,9\}$ is $k$ then $\frac{k}{12}$ is equal to _______
Step-by-Step Solution
Key Concept: Strictly increasing functions correspond bijectively to choosing 6-element subsets from a 9-element set, since the ordering is uniquely determined.
A strictly increasing function $f: \{1,2,3,4,5,6\} o \{1,2,3,...,9\}$ requires $f(1) < f(2) < f(3) < f(4) < f(5) < f(6)$. This is equivalent to choosing 6 distinct elements from the codomain $\{1,2,...,9\}$ and arranging them in increasing order, which can be done in exactly one way once the elements are chosen. Therefore, $k = inom{9}{6} = inom{9}{3} = rac{9 imes 8 imes 7}{3 imes 2 imes 1} = 84$. Thus $ rac{k}{12} = rac{84}{12} = 7$.
Correct Answer: 7