Question

Let $n$ be four digit positive integer in which all the digits are different. If $x$ is number of odd integers and $y$ is number of even integers, then

• A. $x<y$
• B. $x>y$
• C. $x+y=4500$
• D. $|x-y|=56$

Answer 1

Answer: (A), (D)

$|x-y|=56$

When $n$ is odd:
Then units digit can be occupied by $1,3,5,7,9$ and $0$ cannot be on thousands place
The number of such numbers is $x=8\times 8\times 7\times 5=2240.$

When $n$ is even: For any number to be even the last digit should be multiple of $2$ i.e., $0,2,4,6,8$
Case $1:$ If unit's place is filled with $0$
then the total number is $9\times 8\times 7=504.$

Case $2:$ If unit's place is other than $0$,
then the total number is $8\times 8\times 7\times 4=1792.$

Hence, the total number of even numbers is $y=504+1792=2296$