NIMCET 2013 — Mathematics PYQ

1. Understand the Total Range:

Non-negative integers less than $1000$ range from $0$ to $999$.

We can represent all these numbers as $3$-digit numbers (e.g., $0$ as $000$, $5$ as $005$, $25$ as $025$).

Total numbers = $10\times 10\times 10=10^3=1000$.

2. Calculate numbers that do NOT contain the digit $1$:

If a number does not contain the digit $1$, each of the three positions can be filled by any of the remaining $9$ digits $\{0,2,3,4,5,6,7,8,9\}$.

• Digits available = $9$

• Ways to fill 1st position = $9$

• Ways to fill 2nd position = $9$

• Ways to fill 3rd position = $9$

Total numbers without the digit $1$ = $9\times 9\times 9=9^3$.

3. Calculate numbers that contain the digit $1$: