1. We need the number of many-one functions $f:A\to B$ such that $1\in f(A)$, where $A=\{1,2,3,4\},\qquad B=\{1,4,9,16\}.$

Here, many-one means the function is not one-one.

2. First count all functions from $A$ to $B$ such that $1\in f(A)$.

Since $|A|=4$ and $|B|=4$, total number of functions from $A$ to $B$ is $4^4=256.$

Now exclude those functions whose image does not contain $1$. Then each element of $A$ can map only to $\{4,9,16\}$, so number of such functions is $3^4=81.$

Hence, number of functions with $1\in f(A)$ is $256-81=175.$

3. Among these, subtract the one-one functions, because we want many-one functions.

A one-one function from a 4-element set to a 4-element set is actually a bijection. For a bijection to satisfy $1\in f(A)$ is automatic, since every element of $B$ appears exactly once.

Number of one-one functions from $A$ to $B$ is $4!=24.$

4. Therefore, required number of many-one functions is $175-24=151.$

5. Option check:

• A: $151$ ✅
• B: $139$
• C: $163$
• D: $127$

So the correct option is A.