Sets, Relations & Functions
Counting Many-One Functions
nta_pyq_2025_apr
Grade 11
Question:
Let $A = \{1, 2, 3, 4\}$ and $B = \{1, 4, 9, 16\}$. Then the number of many-one functions $f : A \to B$ such that $1 \in f(A)$ is equal to:
Step-by-Step Solution
Key Concept: Total many-one functions = Total functions $-$ one-one functions. Many-one with $1\notin f(A)$ = functions from $A$ to $\{4,9,16\}$ that are many-one = $3^4 - 0$ (no one-one possible from 4 to 3 elements... wait, actually many-one from A to B with $1\notin f(A)$ means all 4 elements map to $\{4,9,16\}$). Then subtract.
Total functions $= 4^4 = 256$. One-one: $4! = 24$. Many-one: $256-24=232$. Many-one with $1\notin f(A)$: all functions to $\{4,9,16\}$ = $3^4=81$ (all many-one). Many-one with $1\in f(A) = 232-81 = 151$.
Correct Answer: 151