Question

Solve ques 3
Q.3. n-digit numbers are formed using only three digits 2, 5 and 7. The smallest value of n for which 900 such distinct numbers can be formed, is
(A) 6
(B) 7
(C) 8
(D) 9

Answer 1

Dear Student,


$$\begin{gathered} IfItisonedigitnumbertherewillbe{3}^{1}suchnumbers(asinthatonedigittherecaneitherbe2,5or7) \\ IfItistwodigitnumbertherewillbe{3}^{2}suchnumbers(asinthatonesplacetherecaneitherbe2,5or7andintensplacetherecaneitherbe2,5or7tooso,3\times 3=3^{2}suchnumbers) \\ ... \\ ... \\ ... \\ IfItisndigitnumbertherewillbe{3}^{n}suchnumbers \\ Now, \\ Accordingtoquestion, \\ {3}^n>900 \\ \Rightarrow {\log }_3{3}^n>{\log }_{3}900 \\ \Rightarrow n{\log }_{3}3>6.1918 \\ \Rightarrow n>6.1918 \\ andnisaninteger \\ so,smallestvalueofnwillneintegerjustafter6.1918,i.e.7 \end{gathered}$$
Regards