The total number of natural numbers of six digits that can

be made with digits 1, 2, 3, 4, if all digits are to appear in the

same number at least once, is 

There can be two types of numbers 

(i) any one of the digits  $1, 2, 3, 4$ repeats thrice and the re

maining  digits only once i.e. of the type $1, 2, 3, 4, 4, 4$

(ii) any two of the digits  $1, 2, 3, 4$ repeat twice and the re

maining two only once i.e. of the type  $1, 2, 3, 3, 4,$

  Number of numbers of the type 1 2 3 4 4 4

       $=\frac{6!}{3!}{\times }^4{C}_1=480$

Number of numbers of the type 1 2 3 3 4 4 

    $=\frac{6!}{2!2!}{\times }^4{C}_2=1080$

  So, the required number $= 480 + 1080 = 1560.$