KA-0023The multiplication principle
4 points, difficulty 2 of 3Level 5–6about 60sA password is one letter from A, B, C followed by two digits from 1 to 5. Digits may repeat. How many passwords are possible?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Count the options at each position.
2The strategy
The three positions are chosen independently, so multiply the number of options at each one. Repeats are allowed, so the digit count does not shrink for the second digit.
3The full solution
3 letters times 5 digits times 5 digits = 75.
Solution— no shortcut on this one
There are three independent positions. The letter has 3 options; the first digit has 5; the second digit also has 5, because repeats are allowed. Multiply: 3 x 5 x 5 = 75. The transferable idea: independent choices multiply, and repeats allowed means the count does not drop for later positions.
Why this is on the test: The multiplication principle is the backbone of contest counting, and adding instead of multiplying is the most common single error in the strand.