CRE 9 - Derangements | Modern Math - Permutation & Combination

Answer: 720

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Explanation :

6 letters can be placed in 6 envelops in 6! = 720 ways.

Hence, 720.

Answer: 1

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Explanation :

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.

Hence, 1.

Answer: 0

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Explanation :

At least two letters must go in wrong envelops.

Hence, 0.

Answer: 0

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Explanation :

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.

Hence, 0.

Answer: 40

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Explanation :

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 $3!\left[1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}\right]$ = 2 ways.

Note: The number of derangements of a set with n objects is given by the formula = $n!\left[1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\dots +\frac{{\left(-1\right)}^n}{n!}\right]$.

∴ Required answer = 20 × 2 = 40 ways.

Hence, 40.

Answer: 135

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Explanation :

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 $4!\left[1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}\right]$ = 9 ways.

∴ Required answer = 15 × 9 = 135 ways.

Hence, 135.