JAMIA 2022 — Mathematics PYQ

Step 1: Total 4-digit numbers possible

The digits available are $\{0,1,2,3,4,5,6,7\}$ (Total 8 digits).

For a 4-digit number:

• Thousands place: Cannot be $0$. So, options are $\{1,2,3,4,5,6,7\}$ (7 options).

• Hundreds place: Any of the 8 digits (8 options).

• Tens place: Any of the 8 digits (8 options).

• Units place: Any of the 8 digits (8 options).

Total 4-digit numbers $=7\times 8\times 8\times 8=7\times 8^3$

Total 4-digit numbers $=7\times 512=3584$

Step 2: 4-digit numbers that do NOT contain the digit 1

To find this, we exclude the digit $1$ from our set. Available digits are $\{0,2,3,4,5,6,7\}$ (Total 7 digits).

For a 4-digit number without $1$:

• Thousands place: Cannot be $0$ or $1$. So, options are $\{2,3,4,5,6,7\}$ (6 options).

• Hundreds place: Any of the 7 digits (7 options).

• Tens place: Any of the 7 digits (7 options).

• Units place: Any of the 7 digits (7 options).

Total numbers without $1=6\times 7\times 7\times 7=6\times 7^3$

Total numbers without $1=6\times 343=2058$

Step 3: Calculation of numbers containing digit 1

$\text{Required Numbers}=(\text{Total 4-digit numbers})-(\text{Numbers without digit 1})$

$\text{Required Numbers}=3584-2058$

$\text{Required Numbers}=1526$

Final Answer:

$1526$