Question
In the decimal system of numeration the number of 6-digit numbers in which the sum of the digits is divisible by 5 is
A.
$$180000$$
B.
$$540000$$
C.
$$5 \times {10^5}$$
D.
None of these
Answer :
$$180000$$
Solution :
Ways: \[\begin{array}{*{20}{c}}
\times \\
9
\end{array}\,\,\begin{array}{*{20}{c}}
\times \\
{10}
\end{array}\,\,\begin{array}{*{20}{c}}
\times \\
{10}
\end{array}\,\,\begin{array}{*{20}{c}}
\times \\
{10}
\end{array}\,\,\begin{array}{*{20}{c}}
\times \\
{10}
\end{array}\,\,\begin{array}{*{20}{c}}
\times \\
6
\end{array}\]
0 cannot go in the first place. In other four places any digit can go. After filling the first five places, the last place can be filled by 0 or 5, 1 or 6, 2 or 7, 3 or 8, 4 or 9 depending upon whether the sum of five digits filled is of the form $$5m, 5m + 4, 5m + 3, 5m + 2$$ or $$5m + 1$$ respectively.
∴ the required number of numbers $$ = 9 \times {10^4} \times 2.$$