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

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Question

(B) We want to find the number of integers between $1$ and $10^{10}$ that contain at least one digit $1$.Total number of integers from $0$ to $10^{10}

Vedclass · Accepted Answer

$10^{10}-9^{10}+1$

Vedclass · Answer

(B) We want to find the number of integers between $1$ and $10^{10}$ that contain at least one digit $1$.Total number of integers from $0$ to $10^{10}-1$ is $10^{10}$.Represent each number as a $10$-digit string (padding with leading zeros,e.g.,$5$ as $0000000005$).The total number of such strings is $10^{10}$.The number of strings that do not contain the digit $1$ is formed by using only the digits ${0, 2, 3, 4, 5, 6, 7, 8, 9}$.There are $9$ choices for each of the $10$ positions,so there are $9^{10}$ such strings.Thus,the number of strings containing at least one $1$ is $10^{10} - 9^{10}$.These strings represent numbers from $0$ to $10^{10}-1$. The string $0000000000$ represents $0$,which does not contain $1$. The string $10000000000$ is not included in our $10$-digit set,but $10^{10}$ itself contains the digit $1$.Since we are looking for numbers between $1$ and $10^{10}$,we exclude $0$ (which doesn't contain $1$) and include $10^{10}$ (which does contain $1$).Therefore,the count is $(10^{10} - 9^{10}) - (	ext{count of } 1	ext{'s in } 0) + (	ext{count of } 1	ext{'s in } 10^{10}) = 10^{10} - 9^{10} - 0 + 1 = 10^{10} - 9^{10} + 1$.

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

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