KA-0143Systematic listing
3 points, difficulty 1 of 3Level 5–6about 45sHow many three-digit numbers can be written using only the digits 1 and 2, with repeats allowed?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Two choices in each of three positions.
2The strategy
Each position independently takes one of two digits, and using a digit does not use it up. Multiply the options position by position rather than adding them.
3The full solution
2 x 2 x 2 = 8. Listing them: 111, 112, 121, 122, 211, 212, 221, 222.
Solution— no shortcut on this one
Each of the three positions can hold either digit, and repeats are allowed so nothing gets used up. That gives 2 x 2 x 2 = 8. Listing confirms it: 111, 112, 121, 122, 211, 212, 221, 222. The transferable idea: independent choices multiply, and the number of positions is an exponent, not a factor to multiply the digits by.
Why this is on the test: Multiplying the digit count by the position count instead of raising it to that power is the standard slip, and choice D is exactly it.