1.) Answer with factorial notations and simplify into numbers.

- 1.1) Find the coefficients of \(x^{5}\) from the expansion of \((2+x+x^{2})^{8}\)
- 1.2) There're 10 people, including A,B,C,D. If we set them arrange in the long bench, find the number of different ways are there, such that only 2 of A,B,C,D are sitting together.

2.) Prove these statements by combinatorial proof.

2.1) \(\displaystyle \sum\limits_{k=0}^{r} \dbinom{r}{k}^{2} = \dbinom{2r}{r}\) for any natural number \(r\).

2.2) \(\displaystyle \frac{(9n)!}{2^{5n}3^{n}}\) is always integer for any natural number \(n\).

3.) Throw 15 6-sided regular dice, find the number of different ways such that every 6 different sides are shown and no more than 3 same sides are shown.

4.) Let \(V(r,n)\) be the number of ways of putting \(r\) different objects into \(n\) identical boxes such that **each boxes must have at least \(k\) objects** Prove that

\[V(r,n) = nV(r-1,n) + \dbinom{r-1}{r-k}V(r-k,n-1)\]

for any natural number \(r,n,k\) and \(r \geq nk\).

5.) There was a rumour inside the group of 10 people. This rumour is spread by e-mail and continuously spread by following rules.

- First, there was only 1 people know about the rumour called
**rumour-er**. - Each e-mail can be either forwarded directly (exactly 1 people and can forward again) or people who receive copies (can be 0 people or any number of people, but cannot forward again)
- People who received the e-mail (by both ways) can know who sent the mail, and those are considered to be
**rumoured** - People who received by forwarding can only forward the mail once and only forward to people who are not
**rumoured**. **Rumour-er**can send as many e-mails as he/she want.

How many ways are there if the e-mails are sent exactly 2 times, and how many ways if sent exactly 3 times?

This is the part of Thailand 1st round math POSN problems.

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TopNewest3) You are looking for the number of solutions to

\( a + b + c + d + e + f = 15 \), subject to \( 1 \leq a \leq 3 \), and similarly for all the other variables.

Hint: Use the substitution \( z = 3 - a \).

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Wow, this is nice! Thank you ^_^

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2.) I use multiset to prove, but I forgot how to form a multiset.

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\(M =\{ 4n\cdot a_{1}, 2n\cdot a_{2}, 2n\cdot a_{3}, 1n\cdot a_{4}\}\) such that \(|M| = 9n\).

Number of permutations = \(\displaystyle \frac{(9n)!}{(4!)^{n}(2!)^{n}(2!)^{n}(1!)^{n}} = \frac{(9n)!}{2^{5n}3^{n}}\) which is always integer. Done!

I swear to goat that I haven't done this for years since the last year test.

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