I have encountered this problem:
A school is planning some trips over the summer. ere are 12 places on the Greece trip, ten places on the China trip and ten places on the Disneyland trip. There are 140 pupils in the school who are all happy to go on any of the three trips. In how many ways can the spaces be allocated?
The textbook has given 140 Choose 12 times 128 Choose 10 times 118 Choose 10, giving an answer of $1.62 * 10^45$.
My question is though, let's say we started with the China trip. That would give us 140 Choose 10 times 130 Choose 12 times 118 Choose 10, which would lead to a different final answer. So how do we know which arrangement is correct? What makes the textbooks arrangement of combinations, 40 Choose 12 times 128 Choose 10 times 118 Choose 10, correct over 140 Choose 10 times 130 Choose 12 times 118 Choose 10?
Thanks in advance for the explanation.
$\frac{140!}{12!128!}*\frac{128!}{10!118!}*\frac{118!}{10!108!}=\frac{140!}{12!10!10!108!}$
$\frac{140!}{10!130!}*\frac{130!}{12!*118!}*\frac{118!}{10!108!}=\frac{140!}{10!12!10!108!}$
So you see they are the same.