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

Answer: (C)

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

Answer 2

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 & one zero as digit ${ }^{9} C _{1}$
Number of 5 -digit numbers containing one non zero and one zero both $=\left(2^{4}-1\right)$
Required prob

$=\frac{\left({ }^{9} C _{2} \times\left(2^{5}-2\right)+{ }^{9} C _{1} \times\left(2^{4}-1\right)\right)}{9 \times 10^{4}}$

$=\frac{36 \times(32-2)+9 \times(16-1)}{9 \times 10^{4}}$

$=\frac{4 \times 30+15}{10^{4}}=\frac{135}{10^{4}}$

$\text{Hence, the correct option is (C): }$ $\frac{135}{10^{4}}$