Question

The number of $4$-digit numbers which can be formed using the digits $0,2,5,7,8$ and that are divisible by 2 is

• A. $300$, when repetition is allowed
• B. $54$, when repetition is not allowed
• C. $60$, when repetition is not allowed
• D. $540$, when repetition is allowed

Answer 1

Answer: (A), (C)

The correct options are
A $300$, when repetition is allowed
C $60$, when repetition is not allowed

Case $1:$ When repetition is not allowed.
For number to be divisible by $2$ the last digit should be $0,2,8$.
If last digit is $0$ then possible numbers $=4\times 3\times 2=24$
If last digit is $2$ or $8$ ,then possible numbers $=3\times 3\times 2\times 2=36$
Hence possible numbers will be
$=36+24=60$

Case $2:$ When repetition is allowed.
First place can be filled by either $2$or $5$ or $7$ or $8$ in $4$ ways
Similarly second and thirdplace can be filled by any one of the given digits $5$ ways
Last place can be filled by either $0$ or $2$ or $8$ in $3$ ways.
$\therefore$The number of $4$ digit numbers divisible by $2$ that can be formed using the digits $0,2,5,7,8$ when repetition is allowed is $=4\times 5\times 5\times 3=300$