Question

The number of all four digits numbers with distinct digits is

• A. $9\times 10\times 10\times 10$
• B. ${}^{10}{P}_4$
• C. $9{\times }^9{P}_3$
• D. none of these

Answer 1

The correct option is C $9{\times }^9{P}_3$

The number of all four digits number with a distinct digit is

There are ten digits 0 to 9

These ten digit have to be put in it places. Thousands, hundreds, tens units

Obviously, 0 cannot take the thousands place because it will become three digit number, so for thousands 9 ways can bedone

Now . and other 8 digit left, so at hundrend place 9 ways to set a digit.

We are now left with 8 digit fill the tens place which can be done in 5 ways

The remaining 7 digit in the unit place in 7 ways

so the answer is $9\times 9\times 8\times 7=4536$

$9\times {{}^{9}P}_3$

$A\to 9$

$B\to 9$

$C\to 8$

$D\to 7$

$=9\times 9\times 8\times 7$