Q. The number of strictly increasing functions \(f\) from the set \(\{1, 2, 3, 4, 5, 6\}\) to the set \(\{1, 2, 3, . . . , 9\}\) such that \(f(i) \neq i\) for \(1 \leq i \leq 6\), is equal to :
Step-by-Step Mathematical Analysis
1. Definitions and Setup:
Domain \(A = \{1, 2, 3, 4, 5, 6\}\) with \(|A| = 6\).
Codomain \(B = \{1, 2, 3, \dots, 9\}\) with \(|B| = 9\).
2. Strictly Increasing Property:
A function is strictly increasing if \(x_1 < x_2 \implies f(x_1) < f(x_2)\). In combinatorics, the number of strictly increasing functions is equivalent to choosing \(r\) distinct elements from a set of \(n\) elements, where \(n=9\) and \(r=6\).
3. Evaluation of Constraint \(f(i) \neq i\):
Since the function is strictly increasing, the following chain must hold:
\(f(1) < f(2) < f(3) < f(4) < f(5) < f(6)\).
For any index \(i\), the minimum possible value \(f(i)\) can take is \(i\).
If we want \(f(i) \neq i\), we must ensure \(f(i) > i\).
4. The Shift Logic:
If \(f(1) > 1\) (meaning \(f(1) \geq 2\)), then:
\(f(2) > f(1) \geq 2 \implies f(2) \geq 3\), so \(f(2) \neq 2\).
\(f(3) > f(2) \geq 3 \implies f(3) \geq 4\), so \(f(3) \neq 3\).
Thus, choosing \(f(1) \geq 2\) automatically ensures \(f(i) \neq i\) for all \(i\).
5. Final Counting:
We need to select 6 elements from the reduced set \(\{2, 3, 4, 5, 6, 7, 8, 9\}\).
The size of this set is \(9 - 1 = 8\).
Total functions = \(^8C_6 = ^8C_2\).
\(^8C_2 = \frac{8 \times 7}{2 \times 1} = 28\).
Deep Dive: Combinatorial Mapping & Functional Analysis
The study of mappings between finite sets is a cornerstone of discrete mathematics. This problem, often categorized under Permutations and Combinations, tests more than just formula application; it tests the understanding of structural constraints. To master these, one must understand the three primary types of monotone mappings.
The "Selection is Arrangement" Principle
One of the most profound realizations in combinatorics is that for strictly monotonic functions (either strictly increasing or strictly decreasing), the act of selecting a subset from the codomain uniquely determines the function. In a standard mapping, we have \(n^r\) possibilities. But when order is enforced, we collapse the permutations into a single valid combination. This is why we use \(^nC_r\). This concept is vital for students aiming for B.Tech Admissions in top-tier Engineering colleges.
Understanding Fixed-Point Constraints in Vectors and Functions
In this problem, the constraint \(f(i) \neq i\) is a Fixed-Point Exclusion. While in derangements (where a set maps to itself) we use the inclusion-exclusion principle, in increasing functions, the inherent order simplifies the task. The inequality chain \(f(1) < f(2) < \dots\) acts as a displacement mechanism. This displacement is frequently used in Algorithm Design to ensure data offsets and memory pointers do not overlap.
Self-Study vs. The Coaching Industry
Traditional JEE Main and Advanced coaching institutes often force students to memorize results for "number of onto functions" or "increasing functions" without explaining the stars and bars method or Pigeonhole Principle applications. By engaging in self-study through detailed digital resources, students develop the analytical depth required to tackle unseen problems. Free educational portals are now bridging the gap, making premium education accessible to every aspirant without the burden of heavy fees.
Advanced Variations: Non-Decreasing Functions
If the question had asked for "non-decreasing" functions (where \(f(x_1) \leq f(x_2)\)), the approach would change entirely. We would allow repetitions, leading to the multiset combination formula: \(^{n+r-1}C_r\). For our set, that would be \(^{9+6-1}C_6 = ^{14}C_6\). Comparing these two scenarios is a favorite tactic in competitive mathematics to filter the top 1% of candidates.
The Logical Bridge to Data Science
Why do we learn this? In Computer Science, strictly increasing functions are used to model indexed arrays and sorted databases. When you ensure \(f(i) \neq i\), you are effectively performing a data shift operation. Mastering these foundations is the first step toward becoming a proficient engineer in fields like Artificial Intelligence and Cybersecurity.
Conclusion for Aspirants
To excel, one must stop looking for shortcuts. True speed comes from clarity. Solve 50 variations of this mapping problem—change the domain size, introduce equality constraints, or change the codomain limits. This deliberate practice is what builds a top-tier rank.
Expert Insights: Frequently Asked Questions
1. Why do we use 8C6 instead of 9C6?
We exclude '1' from the available choices because if we chose 1, it would be forced to map to f(1), violating the condition f(i) != i.
2. Can I use the Derangement formula here?
No. Derangement (D_n) applies to permutations where order is not restricted to being increasing. Here, the 'strictly increasing' rule takes precedence.
3. How many strictly decreasing functions are possible here?
The number would still be based on 9C6, but the constraints f(i) != i would be much harder to satisfy because f(1) would start from the largest value.
4. What is the best way to prepare for JEE Math for free?
Focus on understanding the 'Why' behind every step. Use resources like neetjeerankers.com for step-by-step logic rather than just final answers.
5. What does f(i) != i mean geometrically?
It means the graph of the function never touches the line y = x.
6. Is this question important for NEET Physics?
While the direct math might not appear, the logic of functions and scaling is crucial for Kinematics and Graphs in Physics.
7. How to calculate 14C6 quickly in my head?
Break it down: (14 * 13 * 12 * 11 * 10 * 9) / (6 * 5 * 4 * 3 * 2 * 1). Cancel out the common factors first.
8. What happens if the domain is larger than the codomain?
If |A| > |B|, no strictly increasing function can exist (result = 0).
9. Can strictly increasing functions be periodic?
No, because periodic functions must repeat values, which violates the strictly increasing definition.
10. Where can I find more Permutation and Combination practice?
Check the "Related Questions" section below for word-to-word problems from previous shifts.
Related High-Yield Questions (Must Practice)
Let the mean and variance of 7 observations 2, 4, 10, x, 12, 14, y, x > y, be 8 and 16 respectively. Two numbers are chosen from {1, 2, 3, x - 4, y, 5} one after another without replacement, then the probability, that the smaller number among the two chosen numbers is less than 4, is :
DITCH THE EXPENSIVE COACHING MAFIA! 🛑
Why pay lakhs of rupees for basic concepts? Join the self-study revolution. Get premium, in-depth explanations and thousands of practice questions for absolutely free.
Unlock All Premium Content Now