In writing all of the integers from $$1$$ to $$199$$, how many times is $$'1'$$ used?

Question

Vedantu Content Team · Accepted Answer

Hint: We will take range of $$10$$ numbers and count the digit $$1$$ in that particular range like from $$1 - 10$$, $$11 - 20$$, $$21 - 30$$, and so on keeping all points in mind that $$1$$ appears twice in $$11$$ and in the range $$11 - 20$$, it will appear multiple times. After counting, we will add the number of times $$1$$ appeared to get the result.Complete step by step solution: We have to find the number of times the digit $$1$$ appears in numbers from $$1$$ to $$199$$. From $$1$$ to $$10$$, the digit $$1$$ appears $$2$$ times i.e., in  $$1$$ and $$10$$.From $$11$$ to $$20$$, the digit $$1$$ appears $$10$$ times i.e., in $$11,{\text{ }}12,{\text{ }}13,{\text{ }}14,{\text{ }}15,{\text{ }}16,{\text{ }}17,{\text{ }}18,{\text{ }}19$$ where it appears twice in $$11$$.From $$21$$ to $$30$$, the digit $$1$$ appears only once i.e., in $$21$$.From $$31$$ to $$40$$, the digit $$1$$ appears only once i.e., in $$31$$.From $$41$$ to $$50$$, the digit $$1$$ appears only once i.e., in $$41$$.From $$51$$ to $$60$$, the digit $$1$$ appears only once i.e., in $$51$$.From $$61$$ to $$70$$, the digit $$1$$ appears only once i.e., in $$61$$.From $$71$$ to $$80$$, the digit $$1$$ appears only once i.e., in $$71$$.From $$81$$ to $$90$$, the digit $$1$$ appears only once i.e., in $$81$$.From $$91$$ to $$100$$, the digit $$1$$ appears two times i.e., in $$91$$ and $$100$$.From $$101$$ to $$110$$, the digit $$1$$ appears $$12$$ times i.e., in $$101$$, $$102$$, $$103$$, $$104$$, $$105$$, $$106$$, $$107$$, $$108$$, $$109$$ and $$110$$ where it appears twice in $$101$$ and $$110$$.From $$111$$ to $$120$$, the digit $$1$$ appears $$20$$ times i.e., in $$111$$, $$112$$, $$113$$, $$114$$, $$115$$, $$116$$, $$117$$, $$118$$, $$119$$ and $$120$$ where it appears twice in all except $$111$$ and $$120$$, where it appears thrice and once respectively.From $$121$$ to $$130$$, the digit $$1$$ appears $$11$$ times i.e., in $$121$$, $$122$$, $$123$$, $$124$$, $$125$$, $$126$$, $$127$$, $$128$$, $$129$$ and $$130$$ where it appears twice in $$121$$.From $$131$$ to $$140$$, the digit $$1$$ appears $$11$$ times i.e., in $$131$$, $$132$$, $$133$$, $$134$$, $$135$$, $$136$$, $$137$$, $$138$$, $$139$$ and $$140$$ where it appears twice in $$131$$.From $$141$$ to $$150$$, the digit $$1$$ appears $$11$$ times i.e., in $$141$$, $$142$$, $$143$$, $$144$$, $$145$$, $$146$$, $$147$$, $$148$$, $$149$$ and $$150$$ where it appears twice in $$141$$.From $$151$$ to $$160$$, the digit $$1$$ appears $$11$$ times i.e., in $$151$$, $$152$$, $$153$$, $$154$$, $$155$$, $$156$$, $$157$$, $$158$$, $$159$$ and $$160$$ where it appears twice in $$151$$.From $$161$$ to $$170$$, the digit $$1$$ appears $$11$$ times i.e., in $$161$$, $$162$$, $$163$$, $$164$$, $$165$$, $$166$$, $$167$$, $$168$$, $$169$$ and $$170$$ where it appears twice in $$161$$.From $$171$$ to $$180$$, the digit $$1$$ appears $$11$$ times i.e., in $$171$$, $$172$$, $$173$$, $$174$$, $$175$$, $$176$$, $$177$$, $$178$$, $$179$$ and $$180$$ where it appears twice in $$171$$.From $$181$$ to $$190$$, the digit $$1$$ appears $$11$$ times i.e., in $$181$$, $$182$$, $$183$$, $$184$$, $$185$$, $$186$$, $$187$$, $$188$$, $$189$$ and $$190$$ where it appears twice in $$181$$.From $$191$$ to $$199$$, the digit $$1$$ appears $$10$$ times i.e., in $$191$$, $$192$$, $$193$$, $$194$$, $$195$$, $$196$$, $$197$$, $$198$$ and $$199$$.So, the total number of repetitions of $$1$$ is $$2 + 10 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 2 + 12 + 20 + 11 + 11 + 11 + 11 + 11 + 11 + 11 + 10 = 140$$So, the correct answer is “140”.Note: The counting of a specific digit in a given range of numbers can be done by determining the count of digits on a specific position of numbers in the smaller range and then multiplying it to get the digit in the position for a larger range. This can be repeated for all the positions of the numbers in the given range.

In writing all of the integers from \[1\] to \[199\], how many times is \['1'\] used?