In how many ways can $7$ different books be distributed to $2$ persons if each person gets at least $1$ book?
I did my calculations and my answer is $126$, but the answer stated is $216$.
In how many ways can $7$ different books be distributed to $2$ persons if each person gets at least $1$ book?
I did my calculations and my answer is $126$, but the answer stated is $216$.
There are $7$ different books, which can each be given to one of the two people, leading to $2^7 = 128$ possible solutions. But each person should get at least one book, so the two solutions where one person gets all books should not be counted. So we get $128 - 2 = 126$.
Could it possibly be a typo?
If $i$ denotes the number of books person A receives then $7-i$ is the number of books person B receives. So the number of possibilities is $$\sum_{i=1}^6 \binom{7}{i}=126$$ as you calculated.