Question

The number of $4$-digit natural number which contains

• A. Exactly two distinct digits is $567$
• B. At least two distinct digits is $8990$
• C. At most two distinct digits is $577$
• D. At least three identical digits is $333$.

Answer 1

Answer: (A), (D)

At least three identical digits is $333$.

For two distinct digits :
If '$0$' is not selected $\to {}^9{C}_2\times (2^4-2)$
If '$0$' is selected $\to {}^9{C}_1\times (2^3-1)$
Total $=567$

For at least two digits :
$\to 9\times {10}^3-9=8991$

Atmost two distinct digits :
$\to 567+9=576$

Three identical and one distinct
$\to {}^9{C}_2\times {}^2{C}_1\times 4+{}^9{C}_1\times 4=324$
All identical $\to$ $9$
Total - $333$.