If X={1,2,3,4,5} and Y={a,b,c,d,e,f} and `f:X rarr Y`, find the total number of
`{:((i)" functions ",(ii)" one to one functions "),((iii)" many-one functions ",(iv)" constant functions "),((v)" onto functions ",(vi)" into functions "):}`

Answer

To solve the problem, we need to find the number of different types of functions from set X to set Y, where \( X = \{1, 2, 3, 4, 5\} \) and \( Y = \{a, b, c, d, e, f\} \).

### Step-by-step Solution:

1. **Total Functions**:
The total number of functions from set \( X \) to set \( Y \) is given by \( n^m \), where \( n \) is the number of elements in set \( Y \) and \( m \) is the number of elements in set \( X \).
- Here, \( n = 6 \) (elements in \( Y \)) and \( m = 5 \) (elements in \( X \)).
- Therefore, the total number of functions is:
\[ 6^5 = 7776 \]

2. **One-to-One Functions**:
For one-to-one functions (injective), since \( m < n \), the number of one-to-one functions can be calculated using the formula \( n!/(n-m)! \).
- Here, \( n = 6 \) and \( m = 5 \):
\[ \text{One-to-One Functions} = 6!/(6-5)! = 6! = 720 \]

3. **Many-One Functions**:
Many-one functions can be calculated by subtracting the number of one-to-one functions from the total number of functions.
- Thus, the number of many-one functions is:
\[ \text{Many-One Functions} = \text{Total Functions} - \text{One-to-One Functions} = 7776 - 720 = 7056 \]

4. **Constant Functions**:
The number of constant functions from \( X \) to \( Y \) is equal to the number of elements in \( Y \) since each element in \( X \) must map to the same single element in \( Y \).
- Therefore, the number of constant functions is:
\[ \text{Constant Functions} = n = 6 \]

5. **Onto Functions**:
An onto function (surjective) is not possible when \( m < n \). Therefore, the number of onto functions is:
\[ \text{Onto Functions} = 0 \]

6. **Into Functions**:
Into functions are functions that do not cover the entire codomain. The number of into functions can be calculated as:
\[ \text{Into Functions} = \text{Total Functions} - \text{Onto Functions} = 7776 - 0 = 7776 \]

### Summary of Results:
- (i) Total Functions: **7776**
- (ii) One-to-One Functions: **720**
- (iii) Many-One Functions: **7056**
- (iv) Constant Functions: **6**
- (v) Onto Functions: **0**
- (vi) Into Functions: **7776**