The total number of digits used in numbering the pages of a book having $366$ pages, is

## Concept & Logic To find the total number of digits used to number a book, we must categorize the pages based on how many digits their page numbers have (single-digit, double-digit, triple-digit, etc.) and calculate the digits for each category. ## Step-by-step Solution * **Single-digit pages (1 to 9):** * There are $9$ pages. * Each uses $1$ digit. * Digits used = $9 \times 1 = 9$ digits. * **Double-digit pages (10 to 99):** * Number of pages = $(99 - 10) + 1 = 90$ pages. * Each uses $2$ digits. * Digits used = $90 \times 2 = 180$ digits. * **Triple-digit pages (100 to 366):** * Number of pages = $(366 - 100) + 1 = 267$ pages. * Each uses $3$ digits. * Digits used = $267 \times 3 = 801$ digits. * **Total Digits:** $$ \text{Total} = 9 + 180 + 801 = 990 $$ ## Exam Strategy & Shortcut Memorize the cumulative digit counts for page number problems: * Pages $1-9$ use **9** digits. * Pages $1-99$ use **189** digits. * Pages $1-999$ use **2889** digits. Since $366$ is a $3$-digit number, use the formula: $\text{Total Digits} = 189 + (N - 99) \times 3$ $\text{Total Digits} = 189 + (366 - 99) \times 3$ $\text{Total Digits} = 189 + 267 \times 3 = 189 + 801 = 990$. This is much faster and reduces subtraction errors. ## Common Pitfall The most common mistake is miscounting the number of pages in a range. For example, calculating the number of pages from $100$ to $366$ as $366 - 100 = 266$. You must always add $+1$ when calculating an inclusive range: $(366 - 100) + 1 = 267$. ## Final Answer Therefore, the correct answer is **990**.