Question

A five digit number be selected at random. The probability that the digits in the odd place are odd and the even places are even (repetition is not allowed)

• A. $\frac{{ }^{5} P_{2} \times{ }^{5} P_{3}}{10^{4} \times 9}$
• B. $\frac{{ }^{5} P_{2} \times{ }^{5} P_{3}}{10^{5}}$
• C. $\frac{{ }^{5} C_{2} \times{ }^{5} C_{3}}{10^{4} \times 9} \times 2$
• D. $\frac{{ }^{5} C_{2} \times{ }^{5} C_{3}}{9 \times 10^{4}}$

Answer 1

Answer: (A)

Total number of digits used $ = n = 10$
i , e. $\left\{0, 1, 2, 3. ...9\right\}$
Number of odd digits $ = 5$
Number of even digits $ = 5$
$\overset{\bullet}{odd}\,\,\,\overset{\bullet}{E} \,\,\,\overset{\bullet}{O} \,\,\,\overset{\bullet}{E}\,\,\,\overset{\bullet}{O}$
Now, odd places can be filled by $5P_{3}$ ways. Even places can be filled by $5P_{2}$ ways Required way for five digit number i.e., Number of five digit numbers
$= \overset{\bullet}{\,{}^9C_1} \,\,\,\overset{\bullet}{10\times} \,\,\,\overset{\bullet}{10\times}\,\,\,\overset{\bullet}{10\times}\,\,\,\overset{\bullet}{10}$
$ 9 \times 10^4$
$\therefore $ Required probability $ = \frac{5P_3 \times 5P_2}{10^4 \times 9}$