Of all the mappings that can be defined from the set `A :{1,2,3,4} to B : {5,6,7,8,9}` , a mapping is randomly selected. The chance that the selected mapping is strictly monotonic is

Options

A. `(1)/(125)`
B. `(2)/(125)`
C. `(3)/(25)`
D. `(6)/(25)`

Explanation

To solve the problem of finding the probability that a randomly selected mapping from set A to set B is strictly monotonic, we will follow these steps:

### Step 1: Identify the sets
We have two sets:
- Set A = {1, 2, 3, 4} (which has 4 elements)
- Set B = {5, 6, 7, 8, 9} (which has 5 elements)

### Step 2: Calculate the total number of mappings
Each element in set A can be mapped to any of the 5 elements in set B. Therefore, the total number of mappings (functions) from A to B is calculated as follows:

\[ \text{Total mappings} = 5^{|A|} = 5^4 = 625 \]

### Step 3: Determine the number of strictly monotonic mappings
A mapping is strictly monotonic if it is either strictly increasing or strictly decreasing.

1. **Choosing 4 distinct elements from set B**: Since we need to map 4 elements from A to 4 distinct elements in B, we can choose any 4 elements from the 5 elements in B. The number of ways to choose 4 elements from 5 is given by the combination formula:

\[ \binom{5}{4} = 5 \]

2. **Arranging the chosen elements**: For each selection of 4 distinct elements, there are exactly 2 ways to arrange them in a strictly monotonic manner:
- One arrangement will be strictly increasing.
- The other arrangement will be strictly decreasing.

Thus, for each selection of 4 elements, we have 2 strictly monotonic mappings.

### Step 4: Calculate the total number of strictly monotonic mappings
The total number of strictly monotonic mappings is:

\[ \text{Total strictly monotonic mappings} = \binom{5}{4} \times 2 = 5 \times 2 = 10 \]

### Step 5: Calculate the probability
The probability that a randomly selected mapping is strictly monotonic is given by the ratio of the number of strictly monotonic mappings to the total number of mappings:

\[ \text{Probability} = \frac{\text{Number of strictly monotonic mappings}}{\text{Total mappings}} = \frac{10}{625} \]

To simplify:

\[ \frac{10}{625} = \frac{2}{125} \]

### Final Answer
The probability that the selected mapping is strictly monotonic is:

\[ \frac{2}{125} \]

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