# De-Arrangements

How would one solve this question:

There are 6 friends with 6 different letters E1, E2, E3, E4, E5, E6 to be posted to their respective friends. In how many ways can exactly 3 of the friends receive the right letter? I know that we have to do this problem using de-arrangements but I don't know how to. Please help!!

Note by Siddharth Iyer
5 years, 3 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

There are $$\displaystyle \binom{6}{3}$$ ways to choose the three friends that will get the correct letter. As for the other friends there are two permutations that do not permute any friend to his/her letter (these are permutations with all cycles size greater than 1, which means in this case there is only one cycle size three). Therefore the answer is $\displaystyle \binom{6}{3} \cdot 2 = 40$

- 5 years, 3 months ago

In case we wanted exactly 4 persons, then would the answer be 30?

- 5 years, 3 months ago

Not quite. If you wanted 4 people then there would be $$\displaystyle \binom{6}{2} = 15$$ ways to choose the two people that get the wrong letter. However, there is only one permutation of length 2 that has no cycle of 1, therefore the answer would just be 15.

- 5 years, 3 months ago