Q. 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)

Probability Report Error

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

$\frac{{ }^{5} C_{2} \times{ }^{5} C_{3}}{9 \times 10^{4}}$

Solution:

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}$

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Q. 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)

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