Question 91
There are $24$ pieces each of two different kinds of cakes $A$ and $B$. All the cakes will be distributed among three people $X$, $Y$, $Z$. A method of distribution where a person may not receive any cake of a particular kind is allowed for consideration. How many different ways of distributing cakes are there in which there are no pairs of people ending up with the following situation?
One person in the pair receives $a$ pieces of $A$ and $b$ pieces of $B$, and the other person in the pair receives $a'$ pieces of $A$ and $b'$ pieces of $B$, where all of the conditions $a \le a'$, $b \le b'$ and $a+b \lt a'+b'$ are satisfied.
One person in the pair receives $a$ pieces of $A$ and $b$ pieces of $B$, and the other person in the pair receives $a'$ pieces of $A$ and $b'$ pieces of $B$, where all of the conditions $a \le a'$, $b \le b'$ and $a+b \lt a'+b'$ are satisfied.