Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
If the total number of non-decreasing functions defined from $f : \{1,2,3,4,5\} \to \{1,2,3,4,5,6,7,8,9\}$ is $m$ then $\frac{m}{143}$ is equal to ______.
Step-by-Step Solution
Key Concept: Non-decreasing functions from a finite set to another correspond to multisets, counted by the combination with repetition formula $\binom{n+k-1}{k}$ where $n=9$ and $k=5$.
A non-decreasing function $f : \{1,2,3,4,5\} o \{1,2,3,4,5,6,7,8,9\}$ satisfies $f(1) \leq f(2) \leq f(3) \leq f(4) \leq f(5)$. This is equivalent to choosing a multiset of 5 elements from 9 elements, which equals the number of ways to distribute 5 identical items into 9 distinct bins. Using the stars and bars formula, $m = inom{9+5-1}{5} = inom{13}{5} = rac{13 \cdot 12 \cdot 11 \cdot 10 \cdot 9}{5!} = rac{154440}{120} = 1287$. Therefore, $ rac{m}{143} = rac{1287}{143} = 9$.
Correct Answer: 9