Question

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

• A. $\frac{134}{{10}^4}$
• B. $\frac{121}{{10}^4}$
• C. $\frac{135}{{10}^4}$
• D. $\frac{50}{{10}^4}$

Answer 1

The correct option is C

$\frac{135}{{10}^4}$

The explanation for the correct option.

Total number of cases $=9\times {10}^4$

$9$ for first place and $10$ for rest $4$ places.

Case 1.

Choose two non-zero digits $={}^{9}C_2$

Number of $5-$digit numbers containing both digits $=2^5-2$

$2$ for two cases where all number are same like $55555$

Case 2.

Choose one non-zero and one zero as a digit $={}^{9}C_1$

The number of $5-$digit numbers containing one non-zero and one zero both $=2^4-1$

Favorable case

$$\begin{gathered} ={}^{9}C_2(2^5-2)+{}^{9}C_1(2^4-1) \\ =1080+135 \\ =1215 \end{gathered}$$

The required probability $=\frac{1215}{9\times {10}^4}=\frac{135}{{10}^4}$

Hence, option(C) is the correct answer.