Question

Number of 5 digit numbers divisible by $25$ that can be formed using only the digits $1,2,3,4,5$ and $0$ taken five at a time is

• A. $2$
• B. $32$
• C. $42$
• D. $52$

Answer 1

Answer: (C)

$42$

Given numbers $0,1,2,3,4,5.$ We need to find $5$ digit numbers, since the numbers must be divisible by $25,$ the last $2$ digits must be either $25,50$

$i).$ If last $2$ digits are $2,5$

The first can't be $0$ so only $3$ choices $={}^3{C}_1,$ remaining $2$ places can be filled by the remaining $3$ numbers in ${}^3{P}_2$ ways.

Total $={}^3{C}_1\times {}^3{P}_2=3\times 6=18$

$ii).$ If last $2$ digits are $5,0$

The first $3$ digits can be filled by $4$ numbers in ${}^4{C}_3$ ways.

Total $={}^4{C}_3=24$

$\Rightarrow$ Required $=18+24$

$=42$ numbers possible.

Hence, the answer is $42.$