Question

The total number of $4$ digit numbers that can be formed by using the digits $1,2,3,4,5,6$ and $7$ if at least one digit is repeated is

• A. $7^4$
• B. ${}^7{P}_4$
• C. $7^4-{}^7{P}_4$
• D. ${}^7{P}_4\times 4!$

Answer 1

The correct option is C $7^4-{}^7{P}_4$

The numbers that can be formed when
repetition of digits is allowed are $7^4.$

The numbers that can be formed when
all the digits are distinct or when repetition is not allowed are ${}^7{P}_4$

$\therefore$ The numbers that can be formed when at least one digit is repeated $=7^4-{}^7{P}_4$