If A = {a, b, c, d}, then the number of functions on the set A which are not one-one, is_

Answer

To find the number of functions from the set \( A = \{a, b, c, d\} \) that are not one-one, we will follow these steps:

### Step 1: Calculate the total number of functions from set A to itself.
The formula for the total number of functions from a set with \( n \) elements to itself is \( n^n \). Here, the number of elements in set \( A \) is 4.

\[ \text{Total functions} = 4^4 = 256 \]

### Step 2: Calculate the number of one-one functions from set A to itself.
For a function to be one-one (injective), each element in the domain must map to a unique element in the codomain. Since both the domain and codomain are the same set \( A \), we can calculate the number of one-one functions as follows:

1. The first element can map to any of the 4 elements in \( A \).
2. The second element can map to any of the remaining 3 elements.
3. The third element can map to any of the remaining 2 elements.
4. The fourth element can map to the last remaining element.

Thus, the number of one-one functions is given by:

\[ \text{One-one functions} = 4 \times 3 \times 2 \times 1 = 24 \]

### Step 3: Calculate the number of functions that are not one-one.
To find the number of functions that are not one-one, we subtract the number of one-one functions from the total number of functions:

\[ \text{Not one-one functions} = \text{Total functions} - \text{One-one functions} = 256 - 24 = 232 \]

### Final Answer
The number of functions on the set \( A \) which are not one-one is \( \boxed{232} \).

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