KA-0073Arrangements of a few objects
4 points, difficulty 2 of 3Level 5–6about 60sHow many different four-letter arrangements can be made from the letters of the word MATH, using each letter once?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
How many letters are left for each position?
2The strategy
Fill the positions one at a time. Every letter used is no longer available, so the number of options falls by one at each step.
3The full solution
4 choices for the first letter, then 3, then 2, then 1: 4 x 3 x 2 x 1 = 24.
Solution— no shortcut on this one
Fill the four positions in turn. The first has 4 letters available, the second 3, the third 2, the last 1. Multiplying: 4 x 3 x 2 x 1 = 24. The transferable idea: when items are used up, the count falls by one at each step — that is what makes this different from a code where digits may repeat.
Why this is on the test: It fixes the difference between arranging distinct objects and filling positions that allow repeats, which is the most common mix-up in the strand.