Let A = {1, 2, 3}, B = {a, b, c}, C ` {a_(1), b_(1), c_(1), d_(1), e_(1)}` and consider the following statements.
`S_(1)` : The number of one-one functions from A to C is 60.
`S_(2)` : The number of onto functions from C to A is 150
`S_(3)` : The number of onto functions from B to C is 0
`S_(4)` : The number of objective functions from A to B is 6
Which of the following combination is true ?

Options

A. `S_(1)` and `S_(2)` only
B. `S_(1)` and `S_(3)` only
C. `S_(2)` and `S_(4)` only
D. `S_(1), S_(2), S_(3)` and `S_(4)`

Answer

To solve the problem, we need to evaluate each of the statements \( S_1, S_2, S_3, \) and \( S_4 \) regarding the number of functions between the given sets \( A, B, \) and \( C \).

### Given Sets:
- \( A = \{1, 2, 3\} \) (Cardinality \( |A| = 3 \))
- \( B = \{a, b, c\} \) (Cardinality \( |B| = 3 \))
- \( C = \{a_1, b_1, c_1, d_1, e_1\} \) (Cardinality \( |C| = 5 \))

### Evaluating Each Statement

**Statement \( S_1 \)**: The number of one-one functions from \( A \) to \( C \) is 60.

To find the number of one-one functions from a set of \( m \) elements to a set of \( n \) elements, we use the formula:
\[ nP_m = \frac{n!}{(n-m)!} \]
Here, \( n = 5 \) (elements in \( C \)) and \( m = 3 \) (elements in \( A \)):
\[ 5P3 = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = 5 \times 4 \times 3 = 60 \]
Thus, \( S_1 \) is **True**.

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**Statement \( S_2 \)**: The number of onto functions from \( C \) to \( A \) is 150.

To find the number of onto functions from a set of \( n \) elements to a set of \( m \) elements, we use the formula:
\[ m! \cdot S(n, m) \]
where \( S(n, m) \) is the Stirling number of the second kind, representing the number of ways to partition \( n \) elements into \( m \) non-empty subsets.

Here, \( n = 5 \) and \( m = 3 \). We can calculate \( S(5, 3) \) using the recurrence relation or known values:
\[ S(5, 3) = 25 \]
Thus, the number of onto functions is:
\[ 3! \cdot S(5, 3) = 6 \cdot 25 = 150 \]
So, \( S_2 \) is **True**.

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**Statement \( S_3 \)**: The number of onto functions from \( B \) to \( C \) is 0.

Since \( |B| = 3 \) and \( |C| = 5 \), it is impossible to have an onto function from a smaller set to a larger set. Therefore, the number of onto functions from \( B \) to \( C \) is indeed 0.
Thus, \( S_3 \) is **True**.

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**Statement \( S_4 \)**: The number of bijective functions from \( A \) to \( B \) is 6.

The number of bijective functions (one-to-one and onto) between two sets of equal cardinality \( n \) is given by \( n! \). Here \( n = 3 \):
\[ 3! = 6 \]
Thus, \( S_4 \) is **True**.

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### Conclusion
All statements \( S_1, S_2, S_3, \) and \( S_4 \) are true. Therefore, the correct combination is that all statements are true.

### Final Answer
All statements \( S_1, S_2, S_3, S_4 \) are true.

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