Vivid Colors

Probability Level 3

You have 7 umbrellas, each with a different color, and for every Umbrella you have a sleeve of the same color. In how many ways can you stick the umbrellas into sleeves not of the same color?

Image Credit: Flickr Milica V .

The answer is 1854.

2 solutions

Syed Baqir
Sep 10, 2015

More Precise solution is:

$i!\cdot \sum _{n=0}^7\left(\frac{\left(-1\right)^n}{\:n!}\right)$

$\therefore \quad \rightarrow \quad 7! * \quad \frac{103}{280} = \boxed{1854}$

Vijay Simha
Sep 5, 2015

In combinatorial mathematics, a derangement is a permutation of the elements of a set, such that no element appears in its original position.

The following recurrence relation holds true for derangements:

!n = (n - 1) (!(n-1) + !(n-2)). where !n, known as the subfactorial, represents the number of derangements, with the starting values !0 = 1 and !1 = 0.

We need to find !7 in our case.

Vivid Colors

Image Credit: Flickr Milica V .

The answer is 1854.

2 solutions

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