Question

If a set \( A \) has \( n \) elements, then the number of functions defined from \( A \) to \( A \) that are not one-one is:

• A. \( n^{n^2} \)
• B. \( n! - \left( {^nC_0} + {^nC_1} + {^nC_2} + \cdots + {^nC_n} \right) \)
• C. \( n^n - n! \)
• D. \( n^n \)

Hint:

Remember, total number of functions from a set with \( n \) elements to itself is \( n^n \), and number of one-one functions is \( n! \). Their difference gives the count of not one-one functions.

Answer 1

The Correct Option is C

The total number of functions from a set with \( n \) elements to itself is: \[ n^n \] The number of one-one (injective) functions from \( A \) to \( A \) is: \[ n! \] Therefore, the number of functions that are not one-one is: \[ n^n - n! \]