The right answer is 510 but my calculations keep giving me 252. where did I go wrong? please be thorough because I really do not understand this topic.
2026-04-13 19:46:39.1776109599
Find the number of ways in which 10 different books can be shared between a boy and a girl if each is to receive an even number of books.
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The following solution is assuming that both receive a non zero number of books, i.e, at least two books.
Case 1: Girl gets 8 books, the boy gets 2, so the total combinations will be $$ {10\choose 8} {2\choose 2}=45$$
Case 2: Girl gets 6 books, the boy gets 4, so the total combinations will be $$ {10\choose 6} {4\choose 4}=210$$
Case 3: Girl gets 4 books, the boy gets 6, so the total combinations will be $$ {10\choose 4} {6\choose 6}=210$$
Case 3: Girl gets 2 books, the boy gets 8, so the total combinations will be $$ {10\choose 2} {8\choose 8}=45$$
Add it all up and you get 510.
You can go by this way, but to save time I'd suggest calculating the values in the first two cases and then doubling it.