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
A
`(1)/(125)`
B
`(2)/(125)`
C
`(3)/(25)`
D
`(6)/(25)`
Text Solution
AI Generated Solution
The correct Answer is:
To solve the problem of finding the probability that a randomly selected mapping from set A to set B is strictly monotonic, we can follow these steps:
### Step 1: Determine the total number of mappings from A to B
Set A has 4 elements: {1, 2, 3, 4} and set B has 5 elements: {5, 6, 7, 8, 9}.
The total number of mappings (functions) from set A to set B can be calculated as:
\[
\text{Total mappings} = |B|^{|A|} = 5^4
\]
Calculating this gives:
\[
5^4 = 625
\]
### Step 2: Determine the number of strictly monotonic mappings
A mapping is strictly monotonic if it is either strictly increasing or strictly decreasing.
1. **Strictly Increasing Mappings**: For a mapping to be strictly increasing, we need to choose 4 different elements from set B (since there are 5 elements in B). The number of ways to choose 4 elements from 5 is given by the combination:
\[
\binom{5}{4} = 5
\]
For each selection of 4 elements, there is exactly one way to arrange them in a strictly increasing order.
2. **Strictly Decreasing Mappings**: Similarly, for strictly decreasing mappings, we can also choose 4 different elements from set B. The reasoning is the same as for increasing mappings:
\[
\binom{5}{4} = 5
\]
Again, for each selection of 4 elements, there is exactly one way to arrange them in a strictly decreasing order.
Thus, the total number of strictly monotonic mappings is:
\[
\text{Total strictly monotonic mappings} = 5 \text{ (increasing)} + 5 \text{ (decreasing)} = 10
\]
### Step 3: Calculate the probability of selecting a strictly monotonic mapping
The probability \( P \) 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:
\[
P = \frac{\text{Number of strictly monotonic mappings}}{\text{Total mappings}} = \frac{10}{625}
\]
Simplifying this fraction:
\[
P = \frac{2}{125}
\]
### Final Answer
The chance that the selected mapping is strictly monotonic is:
\[
\frac{2}{125}
\]
To solve the problem of finding the probability that a randomly selected mapping from set A to set B is strictly monotonic, we can follow these steps:
### Step 1: Determine the total number of mappings from A to B
Set A has 4 elements: {1, 2, 3, 4} and set B has 5 elements: {5, 6, 7, 8, 9}.
The total number of mappings (functions) from set A to set B can be calculated as:
\[
\text{Total mappings} = |B|^{|A|} = 5^4
...
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