Thank you for reporting, we will resolve it shortly
Q.
The number of times the digit $3$ will be written when listing the integers from $1$ to $1000$ is
Permutations and Combinations
Solution:
Since $3$ does not occur in $1000$, we have to count the number of times $3$ occurs when we list the integers from $1$ to $999$. Any number between $1$ and $999$ is Of the form $xyz$ where $0 \le x, y z \le9$. Let us first count the numbers in which $3$ occurs exactly once.
Since $3$ can occur at one place in $^{3}C_{1}$ ways, there are $\left(^{3}P_{1}\right)\left(9\times9\right)=3\times9^{2}$ such numbers. Next, 3 can occur in exactly two places in $\left(^{3}C_{1}\right)\left(9\right)=3\times9$ such numbers. Lastly, $3$ can occur in all three digits in one number only. Hence, the number of tithes $3$ occurs is
$1\times\left(3\times9\right)^{2}+2\times\left(3\times9\right)+3\times1$
$=243+54+3=243+57$
$=300$