Question

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

• A. $1560$
• B. $840$
• C. $1080$
• D. $480$

Answer 1

Answer: (A)

$1560$

The total number of six-digit natural numbers that can be made with the digits $1,2,3,4$ if all digits are to appear in the same number at least once is:
We can have $3$ cases:
(1) 3 digits appear once and one digit appears thrice: Select the digit that appears thrice, then arrange the digits: ${}^4{C}_1\times \frac{6!}{3!}=480$
(2) 2 digits appear twice each and the remaining 2 digits appear once each: Select the digits that appear twice, then arrange the digits: ${}^4{C}_2\times \frac{6!}{2!2!}=1080$
Thus, total cases = $480+1080=1560$
Hence, (A) is correct.