Animated Solution for Mathematics - Functions: 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:

First, we must understand the total landscape of mappings from set $A=\{1,2,3,4\}$ to set $B=\{1,4,9,16\}$. Since each of the $4$ elements in $A$ has $4$ possible choices in $B$, the total number of functions is:

This value represents our total universe of possible mappings.

To isolate the many-one functions, we first identify the one-one functions. Since both sets have an equal cardinality of $4$, a one-one function is equivalent to a permutation of the elements of $B$.

The number of one-one functions is given by:

Consequently, the total number of many-one functions is the difference between the total functions and the one-one functions:

The problem requires that the element $1$ must be included in the range of the function. To solve this, we use the complement method: we subtract the number of many-one functions that exclude $1$ from the range from our baseline of $232$.

If $1$ is excluded from the range, the effective codomain becomes $\{4,9,16\}$. The number of functions from $A$ to this reduced codomain is:

We must determine if any of these $81$ functions are one-one. Since we are mapping $4$ elements from the domain into a codomain of only $3$ elements, the Pigeonhole Principle dictates that at least two elements in $A$ must map to the same element in the codomain.

Therefore, all $81$ of these functions are strictly many-one.

To find the number of many-one functions that include $1$ in their range, we subtract the many-one functions that exclude $1$ from the total set of many-one functions:

The total number of many-one functions from $A$ to $B$ such that $1$ is in the range is $151$.