CRE 9 - Derangements | Modern Math - Permutation & Combination
Answer the next 2 questions based on the information given:
There are 6 letters and 6 corresponding envelopes. Each envelope can hold only 1 letter. In how many ways can
The letters be placed into the envelopes?
6 letters can be placed in 6 envelops in 6! = 720 ways.
Each letter be placed into its corresponding envelope?
Each letter can be placed in its corresponding envelope in only 1 way.
6 letters can all be placed in their corresponding envelopes in 16 = 1 way.
Exactly 1 letter be placed into a wrong envelope?
At least two letters must go in wrong envelops.
Exactly 5 letters be placed into corresponding envelopes?
For exactly five letters to be placed in corresponding envelope 1 letter must go in wrong envelope. This is not possible.
This question is same as the previous question.
Exactly 3 letters be placed into wrong envelopes?
For exactly 3 letters to be placed in wrong envelopes 3 letters must go in their corresponding envelopes.
Chose 3 letters and send them in their corresponding envelopes. The number of ways of doing this = 6C3 × 13 = 20 ways.
Now, the remaining letters can all be sent in wrong envelopes in = 2 ways.
Note: The number of derangements of a set with n objects is given by the formula = .
∴ Required answer = 20 × 2 = 40 ways.
Exactly 2 letters be placed into corresponding envelopes?
For exactly 2 letters to be placed in corresponding envelopes 4 letters must go in their wrong envelopes.
Chose 2 letters and send them in their corresponding envelopes. The number of ways of doing this = 6C2 × 12 = 15 ways.
Now, the remaining letters can all be sent in wrong envelopes in = 9 ways.
∴ Required answer = 15 × 9 = 135 ways.
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