Permutations & Combinations
Combinatorial Design
Grade 11
Question:
<p>Eleven criminals want to keep the location of their master criminal in a safe. They want to be able to open the safe only when any 6 of them are present. The safe is thus equipped with a number of different locks, and each criminal is given the keys to some of these locks. What is the minimum number of locks required?</p>
<p>(A) 256</p>
<p>(B) 504</p>
<p>(C) 462</p>
<p>(D) 420</p>
Step-by-Step Solution
Key Concept: Each subset of 5 criminals requires a lock they cannot open; the number of such subsets determines minimum locks needed
<p>For the safe to open when any 6 criminals are present, any group of 5 criminals must be unable to open it. This means for each group of 5 criminals, there must be a lock that none of them can open. The minimum number of locks equals the number of ways to choose 5 criminals from 11, which is \(\binom{11}{5} = 462\).</p>
Correct Answer: C