Question

$n$ digits positive integers formed such that each digit is $1,2$ or $3$. How many of these contain all three of the digits $1,2$ and $3$ atleast once ?

• A. $3(n-1)$
• B. $3^n-{2.2}^n+3$
• C. $3^n-{3.2}^n-3$
• D. $3^n-{3.2}^n+3$

Answer 1

The correct option is D $3^n-{3.2}^n+3$

$n$ digits positive integers formed such that each digit is $1,2$ or $3$.

Therfore total number of integers will be $3^n\to \left(1\right)$.

No. of integers contain only 2 types of digits will be ${}^3{\text{C}}_2{2}^3\to \left(2\right)$

And no. of integers contain only 1 type of digit will be ${}^3{\text{C}}_1{1}^3\to \left(3\right)$

Since (3) are repeated in the (2)

$\therefore 3^n-3\cdot 2^3+3$

Answer