Question

In how many ways can you put 7 letters into their respective envelopes such that exactly 3 go into the right envelope?

Answer 1

Answer-

In 315 ways you can put 7 letters into their respective envelopes such that exactly 3 will go into the right envelope.

Solution

Hint- It's a derangement problem. Derangement is the permutational arrangement with no fixed points. In other words, derangement can be explained as the permutation of the elements of a certain set in a way that no element of that set appears in their original positions.

$\text{It is denoted as}=\ !n\ \text{and calculated as}=\dfrac{n!}{e}$

Where,

n! = the way with all elements appearing in their original position or the permutation of n.

e = Euler's constant.

e.g- The number of ways you put 7 letters, such that no-one gets the right letter is

$!7=\dfrac{7!}{e}=\dfrac{5040}{e}=1854.11\approx1854$

The number of ways can you put 7 letters into their respective envelopes such that exactly 3 go into the right envelope

No of ways in which the 3 correct envelopes can be selected is

$=\binom{7}{3}=35$

Derangement of the remaining 4 envelopes is

$=!4=\dfrac{4!}{e}=\dfrac{24}{e}=8.8\approx9$

Total number of ways of arrangement is,

$=35\times9=315$