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. $\frac{150}{{10}^4}$

Answer 1

The correct option is C $\frac{135}{{10}^4}$

Total cases $=9\left({10}^4\right)$

First case: Choose two non-zero digits $={}^9{C}_2$
Now, number of $5$-digit numbers containing both digits $=2^5-2.$

Second case: Choose one non-zero and one zero as digit $={}^9{C}_1$
Number of $5$-digit numbers containing one non-zero and one zero both $=2^4-1.$

Favourable cases $={}^9{C}_2(2^5-2)+{}^9{C}_1(2^4-1)$
$=1080+135=1215$

Required Probability $=\frac{1215}{9\times {10}^4}=\frac{135}{{10}^4}$