For Father's Day Erica purchased three Father's Day cards. She addressed one envelope to her father, one to her paternal grandfather and one to her maternal grandfather. Accidentally, the cards are randomly inserted into the envelopes and mailed without checking to see if the correct card was placed in the correct envelope. What is the probability that all three cards were placed in the correct envelope?

There are 3 cards which are to be placed in 3 envelopes. We need to find the probability of each card being placed in its correct envelope. For this first we need to find out to total possible number of arrangements of adding 3 cards in the 3 envelopes. Since the order of placing the cards in the envelope matters here, this is a problem of permutations. So we have to find the permutations of 3 objects taken 3 at a time which is represented by 3P3.

The formula for permutations is:

[tex]^{n}P{r}=\frac{n!}{(n-r)!}[/tex]

So, for the given case, it would be:

[tex]^{3}P_{3}=\frac{3!}{(3-3)!}=6[/tex]

This means there are total 6 possible ways of adding the cards to the envelopes. But only 1 correct way of placing the cards in the correct envelopes.

So, the probability that all three cards were placed in the correct envelope = [tex]\frac{1}{6}[/tex]