Question

Eleven scientists are working on a secret project. They wish to lock up the documents in a cabinet such that cabinet can be opened if six or more scientist are present. Then, the smallest number of locks needed is

• A. $460$
• B. $461$
• C. $462$
• D. None of these

Answer 1

Answer: (C)

$462$

For any group of five scientists there must be at least one lock they can not open.

Moreover, for any two different groups of five scientist, there must be two locks they cannot open, because if both cannot open the same lock, there is a group of six scientists among these two groups who will not be able to open the cabinet.

Thus at least ${}^{11}{C}_6=\frac{11\times 10\times 9\times 8\times 7}{1\times 2\times 3\times 4\times 5}=11\times 3\times 2\times 7=462$ locks are needed.