Grinstead-Snell-Probability · Problem 3.1.14
Combinatorics
Mathematics · permutations
Problem
The door on the computer center has a lock which has five buttons numbered from 1 to 5. The combination of numbers that opens the lock is a sequence of five numbers and is reset every week. ((a)) How many combinations are possible if every button must be used once? ((b)) Assume that the lock can also have combinations that require you to push two buttons simultaneously and then the other three one at a time. How many more combinations does this permit?Intuitive explanation
PART (A)
This is about counting ordered arrangements where each button appears exactly once—a permutation problem. Since order matters (pushing 1-2-3-4-5 is different from 5-4-3-2-1), you're arranging 5 distinct objects in 5 positions.
PART (B)
Now you're adding a new type of sequence: push two buttons together (as a single simultaneous step), then three more buttons one at a time. The key insight is that you must count how many ways to choose which 2 buttons go together, then arrange that pair plus the remaining 3 buttons in order. This expands the total number of valid combinations beyond part (a).
Intuitive explanation
PART (A)
This is about counting ordered arrangements where each button appears exactly once—a permutation problem. Since order matters (pushing 1-2-3-4-5 is different from 5-4-3-2-1), you're arranging 5 distinct objects in 5 positions.
PART (B)
Now you're adding a new type of sequence: push two buttons together (as a single simultaneous step), then three more buttons one at a time. The key insight is that you must count how many ways to choose which 2 buttons go together, then arrange that pair plus the remaining 3 buttons in order. This expands the total number of valid combinations beyond part (a).