Mr. A randomly picks 3 distinct numbers from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} and arranges them in descending order to form a three digit number. Mr. B randomly picks 3 distinct numbers from the set {1, 2, 3, 4, 5, 6, 7, 8} and also arranges them in descending order to form a 3 digit number.
Q. The probability that A and B has the same 3 digit number is :

Options

A. `(7)/(9)`
B. `(4)/(9)`
C. `(1)/(84)`
D. `(1)/(72)`

Answer

To solve the problem step by step, we will calculate the probability that Mr. A and Mr. B have the same three-digit number after picking distinct numbers from their respective sets.

### Step 1: Understanding the Sets
- Mr. A picks from the set {1, 2, 3, 4, 5, 6, 7, 8, 9}.
- Mr. B picks from the set {1, 2, 3, 4, 5, 6, 7, 8}.

Since Mr. B does not have the number 9 in his set, for Mr. A and Mr. B to have the same three-digit number, Mr. A must only pick numbers from {1, 2, 3, 4, 5, 6, 7, 8}.

### Step 2: Counting Favorable Outcomes for Mr. A
Mr. A needs to choose 3 distinct numbers from the 8 available numbers (1 to 8). The number of ways to choose 3 numbers from 8 is given by the combination formula:

\[ \text{Number of ways for A} = \binom{8}{3} \]

Calculating this:

\[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]

### Step 3: Counting Total Outcomes for Mr. A
Mr. A can choose any 3 distinct numbers from the 9 available numbers (1 to 9). The total number of ways for Mr. A to choose 3 numbers is:

\[ \text{Total ways for A} = \binom{9}{3} \]

Calculating this:

\[ \binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]

### Step 4: Counting Outcomes for Mr. B
Mr. B also picks 3 distinct numbers from his set of 8 numbers. The number of ways for Mr. B to choose 3 numbers is also:

\[ \text{Number of ways for B} = \binom{8}{3} = 56 \]

### Step 5: Calculating the Probability
The probability that Mr. A and Mr. B have the same three-digit number is given by the ratio of the favorable outcomes for A (which are also valid for B) to the total outcomes for A:

\[ P(\text{same number}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes for A}} = \frac{56}{84} \]

Simplifying this fraction:

\[ P(\text{same number}) = \frac{2}{3} \]

### Step 6: Final Probability Calculation
Since Mr. B has only one way to choose the same numbers that Mr. A has chosen, we need to divide by the total ways Mr. B can choose 3 numbers from his set:

\[ P(\text{same number}) = \frac{1}{\binom{8}{3}} = \frac{1}{56} \]

### Final Answer
Thus, the final probability that Mr. A and Mr. B have the same three-digit number is:

\[ \frac{1}{84} \]