银行保险库的锁与钥匙

“多数”指至少 6 人。

核心约束：任意少于 6 人（即任意 5 人集合）都不能打开金库，因此对每个 5 人集合 $S$，必须存在至少一把锁，使得 $S$ 中无人持有该锁的钥匙。

为使任意加入第 6 人即可打开，最自然的最小构造是：

• 对每个 5 人集合 $S$ 配置一把专属锁；
• 该锁的钥匙发给其补集的 6 人（即不在 $S$ 中的 6 位副总裁）。

这样任意 5 人集合缺少对应那把锁的钥匙而打不开；而任意 6 人集合对任意 5 人集合 $S$ 至少包含一个不在 $S$ 的人，因此 6 人集合总能拿到每把锁的钥匙而打开所有锁。

锁的最少数量至少为 5 人集合的数量：

$$\boxed{\binom{11}{5}=462\text{ 把锁}}.$$

在上述构造中，每把锁需要 6 把钥匙（发给 6 人），所以钥匙总数为

$$\boxed{462\times 6=2772\text{ 把钥匙}}.$$

并且每位副总裁拿到的钥匙数量为

$$\binom{10}{5}=252.$$

---

英文解析

“Majority” means at least 6 people.

Core constraint: Any number of people less than 6 (that is, any group of 5 people) cannot open the vault, so for each group of 5 people $S$, there must be at least one lock so that no one in $S$holds the key to the lock.

In order to open it by joining any 6th person, the most natural minimal construction is:

• Configure an exclusive lock for each group of 5 $S$;
• The key to the lock is given to 6 people (i.e. 6 VPs who are not in $S$).

In this way, any group of 5 people can't open the lock because they lack the key corresponding to that lock; and any group of 6 people $S$ contains at least one person who is not in $S$, so the group of 6 people can always get the key of each lock and open all locks.

Minimum number of locks is the number of collections of at least 5 people:

$$\boxed{\binom{11}{5}=462\text{ locks}}.$$

In the above configuration, each lock requires 6 keys (for 6 people), so the total number of keys is

$$\boxed{462\times 6=2772\text{ keys}}.$$

and the number of keys per VP is

$$\binom{10}{5}=252.$$