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
i)greater than 4000
ii)less than 6000
iii) divisible by 2
iv) divisible by 5
Answered by Rasheed.Sindhi last updated on 30/May/22
$$\bullet\left({i}\right)\:{greater}\:{than}\:\mathrm{4000} \\ $$$${abcd} \\ $$$${a}=\mathrm{4},\mathrm{5},…,\mathrm{9}\:\:\:\:\mathrm{6}\:{ways} \\ $$$${b}….\:\:\:\mathrm{8}\:{ways} \\ $$$${c}…..\:\:\:\mathrm{7}{ways} \\ $$$${d}…….\:\:\:\mathrm{6}{ways} \\ $$$${Total}\:{numbers}:\:\mathrm{6}\centerdot\mathrm{8}\centerdot\mathrm{7}\centerdot\mathrm{6}=\mathrm{2016} \\ $$$$\bullet\:\left({ii}\right)\:{less}\:{than}\:\mathrm{6000} \\ $$$$\:\:\:\:\mathrm{5}\centerdot\mathrm{8}\centerdot\mathrm{7}\centerdot\mathrm{6}=\mathrm{1680}\:{numbers} \\ $$$$\bullet\left({iii}\right)\:{divisible}\:{by}\:\:\mathrm{2} \\ $$$${unit}\:{digit}\:{may}\:{be}\:\mathrm{2},\mathrm{4},\mathrm{6},\mathrm{8} \\ $$$$\mathrm{4}\centerdot\mathrm{8}\centerdot\mathrm{7}\centerdot\mathrm{6}=\mathrm{1344}\:{numbers} \\ $$$$\bullet\left({iv}\right)\:{divisible}\:{by}\:\mathrm{5} \\ $$$${unit}\:{digit}\:{may}\:{be}\:{filled}\:{by}\:\mathrm{5}\:{only} \\ $$$$\mathrm{1}\centerdot\mathrm{8}\centerdot\mathrm{7}\centerdot\mathrm{6}=\mathrm{336}\:{numbers} \\ $$
Commented by MathsFan last updated on 30/May/22
$${thank}\:{you}\:{Sir} \\ $$
Commented by Kledmaniac 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 by Ari last updated on 30/May/22
$${yes}\:{it}\:{is}\:{mistake} \\ $$
Commented by Rasheed.Sindhi last updated on 30/May/22
$$\mathcal{T}{hank}\:{you}\:{sirs}!\:{you}'{re}\:{right}! \\ $$