Question

The different six digit numbers whose 3 digits are even and 3 digits are odd is

• A. 281250
• B. 281200
• C. 156250
• D. none of these

Answer 1

Answer: (A)

281250

If first digit is even then select other places from remaining five places for two more even digits.
This can be $4{\times }^5{C}_2\times 5\times 5.$

Now the remaining three places are to be fit up by odd digit which can be done in $5^3$ ways.
In this way number of six digit number in which three digit are even and $3$ are odd in which three digit are even and $3$ are odd in which the first digit is even
$=4{\times }^5{C}_2\times 5\times 5\times 5^3=8\times 5^6$ (0 can't occurs at first place)
Similarly number of six digit numbers which begin with an odd number/digit and which have $3$ even, $3$ odd digits

$=5\times 5C_2\times 5\times 5\times 5^3=5\times 10\times 5^5={10.5}^6$
$\therefore$ Total six digit number in which $3$ odd $3$ even are

$=8\times 5^6+10\times 5^6=18\times 5^6=281250$