Forest ServicesQ77. 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?
Detailed Solution
To form 4-digit numbers less than 2000 using digits 1, 2, 3, and 4 without repetition, the thousands digit must be 1. The remaining three digits (2, 3, 4) can be arranged in the hundreds, tens, and units places in 3! (3 factorial) ways.
3! = 3 × 2 × 1 = 6 numbers. The numbers are: 1234 1243 1324 1342 1423 1432 Summing these numbers: Units place: 4+3+4+2+3+2 = 18 (write 8, carry 1) Tens place: 3+4+2+4+2+3 + 1 (carry) = 19 (write 9, carry 1) Hundreds place: 2+2+3+3+4+4 + 1 (carry) = 19 (write 9, carry 1) Thousands place: 1+1+1+1+1+1 + 1 (carry) = 7 (write 7) The total sum is 7998.
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