The number of natural numbers less than 1000, | vedclass

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Question

(A) Natural numbers less than $1000$ can be $1$-digit,$2$-digit,or $3$-digit numbers.Case $1$: $1$-digit numbers: The digits can be $1, 2, 3, 4, 5, 6,

Vedclass · Accepted Answer

$738$

Vedclass · Answer

(A) Natural numbers less than $1000$ can be $1$-digit,$2$-digit,or $3$-digit numbers.Case $1$: $1$-digit numbers: The digits can be $1, 2, 3, 4, 5, 6, 7, 8, 9$. Total $= 9$.Case $2$: $2$-digit numbers: The tens place can be filled in $9$ ways (excluding $0$) and the units place can be filled in $9$ ways (including $0$ but excluding the digit used in the tens place). Total $= 9 	imes 9 = 81$.Case $3$: $3$-digit numbers: The hundreds place can be filled in $9$ ways (excluding $0$),the tens place in $9$ ways (including $0$ but excluding the digit used in the hundreds place),and the units place in $8$ ways (excluding the digits used in the hundreds and tens places). Total $= 9 	imes 9 	imes 8 = 648$.Total number of natural numbers $= 9 + 81 + 648 = 738$.

The number of natural numbers less than $1000$,in which no two digits are repeated,is:

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