Question

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

• A. $30$, when repetition is not allowed
• B. $110$, when repetition is not allowed
• C. $160$, when repetition is allowed
• D. $210$, when repetition is allowed

Answer 1

Answer: (A), (C)

$160$, when repetition is allowed

Case $1:$ When repetition is not allowed.
For number to be divisible by $4$, the last two places can be filled by one of the following $08,20,28,52,72,80$ in $6$ ways
Now if the last two digits are $08,20,80$ then first two places can be filled by $2\times 3=6$ ways
If last two digits are $28,52,72$ then first two places can be filled by $2\times 2=4$ ways
Hence total number of ways $=3\times 6+3\times 4=30$

Case $2:$ When repetition is allowed.
For number to be divisible by $4$, the last two places can be filled by one of the following $00,08,20,28,52,72,80,88$ in $8$ ways
The first place can be filled $4$ ways expect $0$
The second place can be filled in $5$ ways.
Total number of ways $=4\times 5\times 8=160$