KEYS

Probability Level 2

Allan, Allen, and Ellen collects keys.
Allan collected 5 keys more than Allen, but Allen gathered 5 keys less than Ellen.
All in all, they have 34 keys.

Allen, who has the least number of keys, decides to put his keys in a key ring.

In how many ways can he arrange his keys in a key ring?

Note: Reflections and rotations are considered identical.


The answer is 2520.

Psalm Lemuel Guevarra by Psalm Lemuel Guevarra , 27, Philippines

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1 solution

Allan has (k-5)+5 keys (simply, k keys), Allen has k-5 keys, and Ellen has k keys. Generating the equation, we have k + (k - 5) + k = 34 Solving for k, we have 8. That means Allen has 8 keys.

When arranged in a key ring, it becomes a "SPECIAL" circular permutation with a formula of (n-1)! divided by two.

So, substituting the given with the formula. Therefore, there are (8-1)! = 5040 / 2 = 2520 ways to arrange the keys in a key ring.

0 Helpful 0 Interesting 0 Brilliant 1 Confused

Let n = allen's keys n + 5 = allan's keys n + 5 = ellen's keys

3n + 10 = 34 -> n = 8

The number of possible arrangements is (8-1)!÷2 = 2520

Joselito Lloren - 7 years, 1 month ago

actually u turned as simple problem into a tricky one by using appropriate words

Prajwal Kavad - 7 years ago

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