Question

How many three-digit numbers are there, with no digit repeated?

Answer 1

Total number of digits = 10
Total number of 3 digit numbers = ${10}_{p_3}$
But these arrangements also include those numbers which have 0 at hundred's place. such numbers are not 3-digit numbers. When 0 is fixed at hundred's place, we have to arrange remaining 9 digits by taking 2 at a time.
The number of such arrangement is $9_{P_3}$.
So, the total of numbers having 0 at hundred's place = $9_{P_2}$
Hence, total number of 3 digit numbers which distinct = ${10}_{P_3}-9_{p_2}$
= $\frac{10!}{(10-3)!}-\frac{9!}{(9-2)!}$
= $\frac{10!}{7!}-\frac{9!}{7!}$
= $\frac{10\times 9\times 8\times 7!}{7!}-\frac{9\times 8\times 7!}{7!}$
= 720-72
= 648.