The total number of seven-digit numbers the s | vedclass

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(B) Let the seven-digit number be represented as $x_1x_2x_3x_4x_5x_6x_7$.The first digit $x_1$ can take any value from $1$ to $9$ ($9$ choices).The di

Vedclass · Accepted Answer

$4500000$

Vedclass · Answer

(B) Let the seven-digit number be represented as $x_1x_2x_3x_4x_5x_6x_7$.The first digit $x_1$ can take any value from $1$ to $9$ ($9$ choices).The digits $x_2, x_3, x_4, x_5, x_6$ can each take any value from $0$ to $9$ ($10$ choices each).For any fixed values of $x_1, x_2, x_3, x_4, x_5, x_6$,the sum $S = x_1 + x_2 + x_3 + x_4 + x_5 + x_6$ is either even or odd.If $S$ is even,then $x_7$ must be even $(0, 2, 4, 6, 8)$ to make the total sum even ($5$ choices).If $S$ is odd,then $x_7$ must be odd $(1, 3, 5, 7, 9)$ to make the total sum even ($5$ choices).In either case,there are exactly $5$ choices for $x_7$ to ensure the sum of all digits is even.Therefore,the total number of such seven-digit numbers is $9 	imes 10 	imes 10 	imes 10 	imes 10 	imes 10 	imes 5 = 4500000$.

The total number of seven-digit numbers the sum of whose digits is even is

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