What is the sum of all 4-digit numbers less than 2000 formed by the digits 1, 2, 3 and 4, where none of the…
What is the sum of all 4-digit numbers less than 2000 formed by the digits 1, 2, 3 and 4, where none of the digits is repeated?
7998
8028
8878
9238
Solution
4-digit numbers less than 2000 using digits 1,2,3,4 without repetition must start with 1. The remaining three positions are filled by 2, 3, 4 in $3! = 6$ ways. Sum of the thousands place $= 6 \times 1000 = 6000$. In each of the other three positions, each of 2, 3, 4 appears $6/3 = 2$ times; digit sum per position $= 2 \times (2+3+4) = 18$. Their contribution $= 18 \times 111 = 1998$. Total $= 6000 + 1998 = 7998$.