Question

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

• A. 1560
• B. 480
• C. 1080
• D. 840

Answer 1

The correct option is A

1560

There can be two types of numbers:
(i) Any one of the digits 1,2,3,4 repeats thrice and the remaining digits only once i.e. of the type 1,2,3,4,4,4 etc.
(ii) Any two of the digits 1,2,3,4 repeats twice and the reamining two only once i.e. of the type 1,2,3,4,4 etc.
Now number of numbers of the (i) type
$=\frac{6!}{3!}\times {}^4{C}_1=480$
Number of numbers of the (ii) type
$=\frac{6!}{2!2!}\times {}^4{C}_2=1080$
Therefore the required number of numbers
=480+1080=1560