`A={1, 2, 3, ..., 9}`. If three numbers are selected from set A and are arranged, the probability that three numbers are either in increasing order or in decreasing order.

Options

A. `1/6`
B. `2/3`
C. `1/2`
D. `1/3`

Answer

To solve the problem, we need to find the probability that three numbers selected from the set \( A = \{1, 2, 3, \ldots, 9\} \) are arranged in either increasing or decreasing order.

### Step-by-Step Solution:

1. **Determine the Total Ways to Select and Arrange 3 Numbers:**
- First, we need to select 3 numbers from the set of 9. The number of ways to choose 3 numbers from 9 is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 9 \) and \( r = 3 \):
\[ \binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]

2. **Calculate the Arrangements of the Selected Numbers:**
- Once we have selected 3 numbers, we can arrange them in \( 3! \) (factorial of 3) ways:
\[ 3! = 6 \]
- Therefore, the total number of arrangements of any 3 selected numbers is:
\[ \text{Total arrangements} = \binom{9}{3} \times 3! = 84 \times 6 = 504 \]

3. **Determine the Favorable Outcomes (Increasing or Decreasing Order):**
- For any set of 3 distinct numbers, there are exactly 2 arrangements that are either in increasing order or in decreasing order. For example, if we select the numbers 1, 2, and 3, they can be arranged as:
- Increasing: \( 1, 2, 3 \)
- Decreasing: \( 3, 2, 1 \)
- Hence, the total number of favorable outcomes is:
\[ \text{Favorable outcomes} = \binom{9}{3} \times 2 = 84 \times 2 = 168 \]

4. **Calculate the Probability:**
- The probability \( P \) that the three selected numbers are either in increasing or decreasing order is given by the ratio of the number of favorable outcomes to the total arrangements:
\[ P = \frac{\text{Favorable outcomes}}{\text{Total arrangements}} = \frac{168}{504} \]
- Simplifying this fraction:
\[ P = \frac{168 \div 168}{504 \div 168} = \frac{1}{3} \]

### Final Answer:
The probability that three numbers selected from set \( A \) are either in increasing order or in decreasing order is \( \frac{1}{3} \).