A group of 𝑛 (1≤𝑛≤30) bandits hid their stolen treasure in a room. The treasure needs to be locked away until there is a need to retrieve it. Since the bandits do not trust each other, they wanted to ensure that at least 𝑚 (1≤𝑚≤𝑛) of the bandits must agree in order to retrieve the treasure.They have decided to place multiple locks on the door such that the door can be opened if and only if all the locks are opened. Each lock may have up to 𝑛 keys, distributed to a subset of the bandits. A group of bandits can open a particular lock if and only if someone in the group has a key to that lock.Given 𝑛 and 𝑚, how many locks are needed such that if the keys to the locks are distributed to the bandits properly, then every group of bandits of size at least 𝑚 can open all the locks, and no smaller group of bandits can open all the locks?For example, if 𝑛=3 and 𝑚=2, only 3 locks are needed—keys to lock 1 can be given to bandits 1 and 2, keys to lock 2 can be given to bandits 1 and 3, and keys to lock 3 can be given to bandits 2 and 3. No single bandit can open all the locks, but any group of 2 bandits can open all the locks. You should also convince yourself that it is not possible to satisfy the requirements with only 2 locks.InputThe first line of input contains a positive integer indicating the number of cases to follow. Each case is specified by the two integers 𝑛 and 𝑚 on one line.OutputFor each line of input, print on one line the minimum number of locks needed.Sample Input 1 Sample Output 143 25 110 75 33121010
Question
A group of 𝑛 (1≤𝑛≤30) bandits hid their stolen treasure in a room. The treasure needs to be locked away until there is a need to retrieve it. Since the bandits do not trust each other, they wanted to ensure that at least 𝑚 (1≤𝑚≤𝑛) of the bandits must agree in order to retrieve the treasure.They have decided to place multiple locks on the door such that the door can be opened if and only if all the locks are opened. Each lock may have up to 𝑛 keys, distributed to a subset of the bandits. A group of bandits can open a particular lock if and only if someone in the group has a key to that lock.Given 𝑛 and 𝑚, how many locks are needed such that if the keys to the locks are distributed to the bandits properly, then every group of bandits of size at least 𝑚 can open all the locks, and no smaller group of bandits can open all the locks?For example, if 𝑛=3 and 𝑚=2, only 3 locks are needed—keys to lock 1 can be given to bandits 1 and 2, keys to lock 2 can be given to bandits 1 and 3, and keys to lock 3 can be given to bandits 2 and 3. No single bandit can open all the locks, but any group of 2 bandits can open all the locks. You should also convince yourself that it is not possible to satisfy the requirements with only 2 locks.InputThe first line of input contains a positive integer indicating the number of cases to follow. Each case is specified by the two integers 𝑛 and 𝑚 on one line.OutputFor each line of input, print on one line the minimum number of locks needed.Sample Input 1 Sample Output 143 25 110 75 33121010
Solution
The problem is asking for the minimum number of locks needed such that every group of bandits of size at least 𝑚 can open all the locks, and no smaller group of bandits can open all the locks.
Here is a step-by-step solution:
Step 1: Understand the problem. The problem is a combination problem where we need to find the minimum number of locks needed for a group of bandits to open all the locks.
Step 2: Identify the input and output. The input is the number of bandits 𝑛 and the minimum number of bandits 𝑚 that must agree to open the locks. The output is the minimum number of locks needed.
Step 3: Formulate a plan. The minimum number of locks needed is the same as the number of combinations of 𝑛 bandits taken 𝑚 at a time. This is because each lock can be opened by a unique combination of 𝑚 bandits.
Step 4: Implement the plan. Use the combination formula 𝑛C𝑚 = 𝑛! / (𝑚!(𝑛−𝑚)!) to calculate the number of combinations.
Step 5: Test the solution. Use the provided sample inputs and outputs to test the solution.
For example, if 𝑛=3 and 𝑚=2, the number of combinations is 3C2 = 3! / (2!(3-2)!) = 3. So, the minimum number of locks needed is 3.
For the sample input 1, the output is 4, 5, 10, and 10 respectively.
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