Question

The number of natural numbers less than $1000$ that are divisible by $5$ in which no digit occurs more than once in the same number is

• A. 154
• B. 136
• C. 144
• D. 152

Answer 1

Answer: (A)

Digits are $0,1,2,3,4,5,6,7,8,9$

Case 1:
Number of $3$ digit numbers that end with $5$

$(–,–,5)$

For first blank $=8_{P_1}=8$ [Since, $0$ can’t be first place]

For second blank $=8_{P_1}=8$ [Since, out of $10$ numbers two numbers already filled]

Number of $3$ digit numbers that end with $5=8\times 8=64$

Case 2:

Number of $3$ digit numbers that end with $0$

$(–,–,0)$

For first blank $=9_{P_1}=9$

For second blank $=8_{P_1}=8$ [Since, out of $10$ numbers two numbers already filled]

Number of $3$ digit numbers that end with $0=9\times 8=72$

Case 3:
Number of 2 digit numbers that end with $5$

$(–,5)$

In blank $=8_{P_1}=8$ [Since, $0$ can’t be first place]

Case 4:
Number of $2$ digit numbers that end with $0$

$(–,0)$

In blank $=9_{P_1}=9$

Case 5:
Single digit number that ends with $5$ is $1$.
From all five cases,

Total number of ways $=64+72+8+9+1=154$

Hence, Option A is correct.