The number of integers between 1 and 10^{10} | vedclass

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Question

(A) We are looking for the number of integers in the set ${1, 2, 3, ..., 10^{10}}$ that contain at least one digit $1$.It is easier to calculate the c

Vedclass · Accepted Answer

$10^{10}-9^{10}$

Vedclass · Answer

(A) We are looking for the number of integers in the set ${1, 2, 3, ..., 10^{10}}$ that contain at least one digit $1$.It is easier to calculate the complement: the number of integers that do not contain the digit $1$.Consider all numbers from $0$ to $10^{10}-1$. Each of these can be represented as a $10$-digit string (padding with leading zeros if necessary) using the digits ${0, 1, 2, ...., 9}$.There are $10^{10}$ such strings in total.The number of strings that do not contain the digit $1$ is formed by choosing each of the $10$ positions from the remaining $9$ digits ${0, 2, 3, ...., 9}$.Thus,there are $9^{10}$ such strings.One of these strings is $0000000000$,which corresponds to the number $0$. Since we are looking for integers between $1$ and $10^{10}$,we exclude $0$.So,the number of integers between $1$ and $10^{10}-1$ that do not contain the digit $1$ is $9^{10}-1$.The total number of integers between $1$ and $10^{10}-1$ is $10^{10}-1$.Therefore,the number of integers between $1$ and $10^{10}-1$ that contain at least one $1$ is $(10^{10}-1) - (9^{10}-1) = 10^{10}-9^{10}$.Finally,we check the number $10^{10}$. It does not contain the digit $1$. Thus,the count remains $10^{10}-9^{10}$.

The number of integers between $1$ and $10^{10}$ (inclusive) that contain the digit $1$ is:

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