Question

Show the total number of natural numbers of six digit that can be made with digits. $1,2,3,4,$ if all numbers are to appear in the same number at least once is $1560$.

Answer 1

We have to choose numbers of 6 digits out of given 4 digits using all the four. This possible some of the digits repeat to make 6 digits.
I. $3$ different, $3$ alike $(1,1,1,2,3,4)$
Only one number out of four will appear three times. This can be done in ${}^4{C}_1=4$ ways.
Now we have a set of 6 digits out of which three are alike and they can be arranged in $\frac{6!}{3!\left(alike\right)}=\mathrm{6.5.4}=120$
hence by fundamental theorem the number of such numbers $=4\times 120=480$.
II. $2$ alike, $2$ alike, $2$ different $(1,1,2,2,3,4)$ out of $4$ digits we can select $2$ sets of alike in ${}^4{C}_2=\frac{4.3}{1.2}=6$ ways.
Now we have a set of 6 digits out of which 2 are alike of one kind and 2 of other kind. They can be arranged in $\frac{6!}{2!2!\left(alike\right)}=\frac{720}{4}=180$ ways.
Hence by fundamental theorem the number of such numbers $=6\times 180=1800$ ways
Total $=480+1080=1560$.