How many numbers greater than 1000 but not gr | vedclass

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Question

(B) The numbers must be $4$-digit numbers between $1001$ and $4000$ inclusive.For a $4$-digit number $d_1 d_2 d_3 d_4$,the first digit $d_1$ can be $1

Vedclass · Accepted Answer

$375$

Vedclass · Answer

(B) The numbers must be $4$-digit numbers between $1001$ and $4000$ inclusive.For a $4$-digit number $d_1 d_2 d_3 d_4$,the first digit $d_1$ can be $1, 2, 3$ or $4$.If $d_1 \in \{1, 2, 3\}$,then $d_2, d_3, d_4$ can each be any of the $5$ digits $\{0, 1, 2, 3, 4\}$.Number of such integers = $3 	imes 5 	imes 5 	imes 5 = 375$.However,this includes $1000$ (which is not greater than $1000$) and includes $4000$ (which is allowed).Wait,the range is $1000 If $d_1 = 1, 2, 3$,there are $3 	imes 5^3 = 375$ numbers. This set includes $1000$ (when $d_1=1, d_2=0, d_3=0, d_4=0$).Since we need numbers $> 1000$,we exclude $1000$,so $375 - 1 = 374$.If $d_1 = 4$,the only number $\le 4000$ is $4000$ itself.Total numbers = $374 + 1 = 375$.

How many numbers greater than $1000$ but not greater than $4000$ can be formed using the digits $0, 1, 2, 3, 4$ if repetition of digits is allowed?

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