Sets And Functions — Medium JEE math MCQ
**[JEE Mains 2026]** If A = {1, 2, 3, 4, 5, 6} and B = {1, 2, 3, ..., 9}, then the number of strictly increasing functions f: A → B such that f(x) ≥ x for all x ∈ A is:
- A. 56
- B. 84
- C. 126
- D. 28
Solution
For a strictly increasing function f: A → B with f(x) ≥ x:
f(1) ≥ 1, f(2) > f(1) and f(2) ≥ 2, ..., f(6) ≥ 6.
We need to choose 6 values from {1, 2, ..., 9} such that the ith chosen value ≥ i.
Let g(i) = f(i) - i + 1. Then g(1), g(2), ..., g(6) are chosen from adjusted sets.
This is equivalent to choosing 6 elements from {1, 2, ..., 9} maintaining certain conditions.
Number of ways = C(9, 6) = 84 after accounting for the constraint.