Question

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

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

Answer 1

The correct option is C $462$

For any group of $5$ scientists there must at least be one lock they cannot open.

Moreover for any two different groups of $5$ scientists, they must be two locks they cannot open because if both groups cannot open the same lock, there is a group of 6 scientists among these $2$ groups which will not be able to open the cabinet.

Thus atleast,

${=}^{11}{C}_6=11\times 3\times 2\times 7=462$ locks are needed.