Question

How many numbers between $400$ and $1000$ can be formed with the digits $0,2,3,4,5,6,$ if no digit is repeated in the same number?

Answer 1

Number between $400$ and $1000$ consist of three digits with digit at hundred’s place greater than or equal to $4$. Hundred’s place can be filled, by using the digits $4,5,6$ in $3$ ways. Now, ten’s and unit’s places can be filled by the remaining $5$ digits in ${}^5{P}_2$ ways.

Hence, the required number of numbers $=3{\times }^5{P}_2=3\times \frac{5!}{3!}=3\times \frac{5\times 4\times 3!}{3!}=3\times 20=60$