Alice was so bored, waiting for a message from Bob, that she started to play with the five white rabbits she had got from the Queen of Hearts. She tried to figure out in how many ways she could split her rabbits in groups, like 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1, so, for 5 rabbits, 7 ways.
She decided to count the partitions in which no group contained more than 3 rabbits… in our example, there are 5. And then, she counted the partitions with no more than 3 groups. Amazingly, although the groups were not the same, the two numbers coincided, also 5.
Alice wondered… She wonders all the time (why?). Is that a coincidence?