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How many $4$-digit numbers, each greater than $1000$ and each having all four digits distinct, are there with $7$ coming before $3$

Four digit number can be from such that each greater than $1000.$

And $7$ comming before $3.$

$\text{Case}1: \; 7, \; 3, \; 8\text{ways}, \; 7\text{ways} = 8 \times 7 = 56$

$7, 8 \text{ways}, 3, 7\text{ways} = 8 \times 7 = 56$

In case$1,$ total number of ways $= 3 \times 56 = 168 \; \text{ways.}$

$\text{Case}2 :$

In case$2,$ total number of ways $= 2 \times 49 = 98 \; \text{ways}.$

$\text{Case}3 :$

Thus, total such four digit numbers $= 168 + 98 + 49 = 315.$

$\therefore$ The $4 – \text{digit}$ number greater than $1000$ with $7$ before $3$ is $415.$

Correct Answer$: 315$
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