Question

${\text{Show the total number of natural numbers of six digit that can be made with }} \\\ {\text{digits}}{\text{. }}1,2,3,4,{\text{ if all numbers are to appear in the same number at least once in 150,}} \\\$

Answer 1

$${\text{We have to choose numbers having }}6{\text{ digits and made using all the four digits 1,2,3,4}}{\text{. }} \\\ {\text{This is possible if some of the digits repeat to make }}6{\text{ digits}}. \\\ {\text{So, there can be two cases,}} \\\ {\text{Case }}1:{\text{ Out of 6 digits }}3{\text{ are different}},\;3{\text{ are }}\;{\text{same}}\; \\\ {\text{ This case occurs only when one number out of four will appear three times}}{\text{.}} \\\ {\text{ So, for selecting one number that occurs 3 times in 6 digit number is,}} \\\ {\text{ This can be done in}}\;{\text{ }}{}^4{C_1}ways = 4{\text{ }}ways. \\\ {\text{ Now we have a set of }}6{\text{ digits out of which three are same and they can be arranged in}}\;{\text{ }}\dfrac{{6!}}{{3!(same)}} = 120ways. \\\ {\text{ Hence by fundamental theorem the number of such numbers\; = 4 \times 120 = 480}}{\text{.}} \\\ {\text{Case 2}}:{\text{ Out of 6 digits 2 are\;same}},2{\text{ are same,}}\;2{\text{ are different }}\; \\\ {\text{ This case occurs only when out of the 4 digits we can select 2 sets of same }}{\text{.}} \\\ {\text{ So, for selecting 2 number from 4 dgits that occurs 2 times in 6 digit number is }} \\\ {\text{ This can be done in}}\;{\text{ }}{}^4{C_2}ways = 6{\text{ }}ways. \\\ {\text{ Now we have a set of }}6{\text{ digits out of which 2 are same of one kind and 2 of other kind}}{\text{.}} \\\ {\text{ They can be arranged in}}\;{\text{ }}\dfrac{{6!}}{{2!(same)2!(same)}} = \dfrac{{720}}{4} = 180ways. \\\ {\text{ Hence by fundamental theorem the number of such numbers\; = 6 \times 180 = 1880 }}{\text{.}} \\\ {\text{So total number of numbers that can be made from both case will be 480 + 1080 = 1560}}{\text{.}} \\\ {\text{Hence proved}} \\\ {\text{NOTE: - Whenever you came up with this type of problem then the best way is to break the }} \\\ {\text{problem into different cases and find the value for each case then add them up}}{\text{.}} \\\ \\\$$