CAT 2025 Slot3 Quant: The sum of all the digits of the number

Step 1: Analyze the second part $(10^{25} - 123)$
$10^{25}$ is 1 followed by 25 zeros. Subtracting 123:
$100,000...000 - 123 = 99,999...99877$.
Since we subtract from the $25^{th}$ power, the resulting number has 25 digits total.
The last three digits are $877$ (from $1000 - 123$). The remaining $25 - 3 = 22$ digits are all $9$s.

Step 2: Combine with $10^{50}$
$10^{50}$ is 1 followed by 50 zeros. The number $10^{25}-123$ will occupy the last 25 places of these zeros.
The number looks like:
$1 \underbrace{00...00}_{25 \text{ zeros}} \underbrace{99...99}_{22 \text{ nines}} 877$

Step 3: Calculate Sum of Digits
Sum $= 1 (\text{from } 10^{50}) + 0 (\text{zeros}) + (22 \times 9) + 8 + 7 + 7$
Sum $= 1 + 198 + 22$
Sum $= \mathbf{221}$.