Let A={1,2,3,4,5} and B={-2,-1,0,1,2,3,4,5}.
Increasing function from A to B is

Options

A. 120
B. 72
C. 60
D. 56

Answer

To find the number of increasing functions from set A to set B, we can follow these steps:

### Step 1: Identify the sets
Let \( A = \{1, 2, 3, 4, 5\} \) and \( B = \{-2, -1, 0, 1, 2, 3, 4, 5\} \).

### Step 2: Determine the number of elements in each set
Count the elements in each set:
- The number of elements in set \( A \) is \( |A| = 5 \).
- The number of elements in set \( B \) is \( |B| = 8 \).

### Step 3: Choose elements from set B
Since we need to create an increasing function from \( A \) to \( B \), we must choose 5 distinct elements from set \( B \). The order of selection does not matter because the function must be increasing.

### Step 4: Calculate the combinations
We need to calculate the number of ways to choose 5 elements from 8 elements in set \( B \). This is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of elements to choose from, and \( r \) is the number of elements to choose.

In our case, \( n = 8 \) and \( r = 5 \):
\[ \binom{8}{5} = \frac{8!}{5!(8-5)!} = \frac{8!}{5! \cdot 3!} \]

### Step 5: Simplify the combination
Calculating \( \binom{8}{5} \):
\[ \binom{8}{5} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]

### Final Answer
Thus, the number of increasing functions from set \( A \) to set \( B \) is \( 56 \).

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