Question

The probability that a randomly chosen 5 -digit number is made from exactly two digits is :

• A. $\frac{121}{10^{4}}$
• B. $\frac{150}{10^{4}}$
• C. $\frac{135}{10^{4}}$
• D. $\frac{134}{10^{4}}$

Answer 1

The Correct Option is C

To find the probability that a randomly chosen 5-digit number is made from exactly two different digits, we will approach the problem step-by-step. 

1. Understanding Possible Combinations:
   • A 5-digit number can be formed using digits 0-9, but the first digit cannot be 0 (as it wouldn't be a 5-digit number).
   • Since the number should be composed of exactly two different digits, let us choose any 2 distinct digits from the set {0, 1, 2, ..., 9}. There are \(^9C_2\) ways to select these two digits.
2. Calculating the Total Number of Combinations:
   • The first digit should be non-zero. Hence, if 0 is one of the chosen digits, it cannot be the leading digit, leaving only the other digit (1-9) as a possibility for the first position.
   • If both digits are non-zero, then either can be placed in the first position.
   • Using these rules, calculate the total combinations.
3. Calculating Probabilities for Different Cases:
   • Case 1: 0 is one of the chosen digits. Choose another digit from 1 to 9.
     • Number of ways to choose 2 digits: \((^1C_1 \cdot ^9C_1) = 9\).
     • The non-zero digit must occupy the first position; remaining 4 positions can be any combination of the two digits.
     • Total ways for these 4 positions: \(2^4 = 16\), so total for this arrangement: \(9 \cdot 16 = 144\).
   • Case 2: Both digits are non-zero.
     • Number of ways to choose 2 non-zero digits: \(^9C_2 = 36\).
     • Either digit can occupy the first position.
     • Total for remaining positions: \(2^4 = 16\), so for all possible first positions: \(36 \cdot 16 = 576\).
4. Calculating Total Probability:
   • Add the results: \(144 + 576 = 720\) numbers use exactly two digits total.
   • Total possible 5-digit numbers (10000 to 99999): \(90000\).
   • Hence, the probability is: \(\frac{720}{90000} = \frac{135}{10000}\).
5. Conclusion:

The probability that a randomly chosen 5-digit number is made from exactly two digits is \(\frac{135}{10000}\), which matches with the correct option given.