First slide

Combinations

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 scientists are present. Then, the smallest number of locks needed is

Moderate

A

460
B

461
C

462
D

None of these

Solution

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 scientists, there must be two locks they cannot open, because if both groups 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
$\begin{matrix} ^{11} C_{6} = \frac{11 \times 10 \times 9 \times 8 \times 7}{1 \times 2 \times 3 \times 4 \times 5} = 11 \times 3 \times 2 \times 7 \\ = 462 locks are needed \end{matrix}$

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