Total number less than 3×108 and can be formed using the digits 1,2,3 is equal to

Formed number can be utmost of nine digits. Total number of such numbers is3+32+33+⋯+38+2×38=338−13−1+2×38=39−3+4×382=7×38−32

Total number less than 3×108 and can be formed using the digits 1,2,3 is equal to

Formed number can be utmost of nine digits. Total number of such numbers is3+32+33+⋯+38+2×38=338−13−1+2×38=39−3+4×382=7×38−32

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Total number less than $3 \times 10^{8}$ and can be formed using the digits 1,2,3 is equal to

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$\frac{1}{2} (3^{9} + 4 \times 3^{8})$

$\frac{1}{2} (3^{9} - 3)$

$\frac{1}{2} (7 \times 3^{8} - 3)$

$\frac{1}{2} (3^{9} - 3 + 3^{8})$

answer is C.

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Detailed Solution

Formed number can be utmost of nine digits. Total number of such numbers is
$\begin{array}{l} 3 + 3^{2} + 3^{3} + \dots + 3^{8} + 2 \times 3^{8} \\ = \frac{3 (3^{8} - 1)}{3 - 1} + 2 \times 3^{8} = \frac{3^{9} - 3 + 4 \times 3^{8}}{2} = \frac{7 \times 3^{8} - 3}{2} \end{array}$