Question Number 170748 by MathsFan last updated on 30/May/22
$$ \\ $$ A number of four different digits is to be formed from the digits 1,2,3,4,5,6,7,8,9.Find how many of them are: i)greater than 4000 ii)less than 6000 iii) divisible by 2 iv) divisible by 5\\n
Answered by Rasheed.Sindhi last updated on 30/May/22
$$\bullet\left({i}\right)\:{greater}\:{than}\:\mathrm{4000} \\ $$ $${abcd} \\ $$ $${a}=\mathrm{4},\mathrm{5},...,\mathrm{9}\:\:\:\:\mathrm{6}\:{ways} \\ $$ $${b}....\:\:\:\mathrm{8}\:{ways} \\ $$ $${c}.....\:\:\:\mathrm{7}{ways} \\ $$ $${d}.......\:\:\:\mathrm{6}{ways} \\ $$ $${Total}\:{numbers}:\:\mathrm{6}\centerdot\mathrm{8}\centerdot\mathrm{7}\centerdot\mathrm{6}=\mathrm{2016} \\ $$ $$\bullet\:\left({ii}\right)\:{less}\:{than}\:\mathrm{6000} \\ $$ $$\:\:\:\:\mathrm{\color{mathred}{5}}\centerdot\mathrm{8}\centerdot\mathrm{7}\centerdot\mathrm{6}=\mathrm{\color{mathred}{1}\color{mathred}{6}\color{mathred}{8}\color{mathred}{0}}\:{numbers} \\ $$ $$\bullet\left({iii}\right)\:{divisible}\:{by}\:\:\mathrm{2} \\ $$ $${unit}\:{digit}\:{may}\:{be}\:\mathrm{2},\mathrm{4},\mathrm{6},\mathrm{8} \\ $$ $$\mathrm{4}\centerdot\mathrm{8}\centerdot\mathrm{7}\centerdot\mathrm{6}=\mathrm{1344}\:{numbers} \\ $$ $$\bullet\left({iv}\right)\:{divisible}\:{by}\:\mathrm{5} \\ $$ $${unit}\:{digit}\:{may}\:{be}\:{filled}\:{by}\:\mathrm{5}\:{only} \\ $$ $$\mathrm{1}\centerdot\mathrm{8}\centerdot\mathrm{7}\centerdot\mathrm{6}=\mathrm{336}\:{numbers} \\ $$
Commented byMathsFan last updated on 30/May/22
$${thank}\:{you}\:{Sir} \\ $$
Commented byKledmaniac last updated on 30/May/22
$$ \\ $$ $${i}\:{think}\:{there}'{s}\:{a}\:{mistake}\:{in}\:\left({ii}\right) \\ $$ $${it}\:{should}\:{be}\:\mathrm{5}\centerdot\mathrm{8}\centerdot\mathrm{7}\centerdot\mathrm{6}=\mathrm{1680} \\ $$
Commented byAri last updated on 30/May/22
$${yes}\:{it}\:{is}\:{mistake} \\ $$
Commented byRasheed.Sindhi last updated on 30/May/22
$$\mathcal{T}{hank}\:{you}\:{sirs}!\:{you}'{re}\:{right}! \\ $$