Question

How many numbers can be formed by using any number of the digits 3, 1, 0, 5, 7, 2, 9, no digit being repeated in any number?

Answer 1

The number of single digit numbers is ⁷P₁. 

The permutations of 7 digits taken 2 at a time are ⁷P₂. But ⁶P₁ of these have zero in the ten’s place and so reduce to one digit numbers. 

Hence the number of two-digit numbers is ⁷P₂ - ⁶P₁ 

Similarly the number of the three-digit numbers is ⁷P₃ - ⁶P₂  and so on. 

∴ The total number required 

= ⁷P_l + (⁷P₂ - ⁶P₁ ) + (⁷P₃ - ⁶P₂) + (⁷P₄ - ⁶P₃) + (⁷P₅ - ⁶P₄) + (⁷P₆ - ⁶P₅) + (⁷P₇ - ⁶P₆) = 11743.