Let A={1,2,3,4,5} and B={-2,-1,0,1,2,3,4,5}.
Non-decreasing functions from A to B is

Options

A. 216
B. 540
C. 792
D. 840

Answer

To find the number of non-decreasing functions from set A to set B, we can use the combinatorial method. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Identify the Sets**:
- Let \( A = \{1, 2, 3, 4, 5\} \) which has \( m = 5 \) elements.
- Let \( B = \{-2, -1, 0, 1, 2, 3, 4, 5\} \) which has \( n = 8 \) elements.

2. **Understanding Non-Decreasing Functions**:
- A function \( f: A \to B \) is non-decreasing if for any \( x_1, x_2 \in A \) such that \( x_1 < x_2 \), we have \( f(x_1) \leq f(x_2) \).

3. **Using the Combinatorial Formula**:
- The number of non-decreasing functions from a set with \( m \) elements to a set with \( n \) elements can be calculated using the formula:
\[ \binom{m+n-1}{m} \]
- Here, \( m = 5 \) and \( n = 8 \).

4. **Substituting Values into the Formula**:
- Calculate \( m+n-1 \):
\[ m+n-1 = 5 + 8 - 1 = 12 \]
- Now, we need to compute \( \binom{12}{5} \).

5. **Calculating \( \binom{12}{5} \)**:
- The formula for combinations is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
- Thus,
\[ \binom{12}{5} = \frac{12!}{5!(12-5)!} = \frac{12!}{5! \cdot 7!} \]
- This simplifies to:
\[ \binom{12}{5} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} \]

6. **Performing the Calculation**:
- Calculate the numerator:
\[ 12 \times 11 = 132 \]
\[ 132 \times 10 = 1320 \]
\[ 1320 \times 9 = 11880 \]
\[ 11880 \times 8 = 95040 \]
- Calculate the denominator:
\[ 5! = 120 \]
- Now divide:
\[ \frac{95040}{120} = 792 \]

7. **Final Answer**:
- The number of non-decreasing functions from set A to set B is \( 792 \).