Combinatorial Problem

Its diwali today and right now I am in the library. So far it was good today, I went out to have lunch at the local baptist church which does the Tuesday afternoon lunch. On my homework front I did few more problems. There is this interesting problem of (1+sqrt(2)+sqrt(3))^(8) and you have to simplify it. The multinomial theorem seems to be the best option but still there is enough calculation which makes me wonder if there is a shorter way. But still if you look at the original problem and do it by brute force you will end up with 3^8= 6561 operations and with multinomial theorem its easy to see that the number of terms will be comb(8+2,2) = 45 which is a huge advantage. To find the individual terms as one can look at it that there will be 4 different terms in the form a+b*sqrt(2)+c*sqrt(3)+d*sqrt(6). To find constant "a" takes 15 terms and the rest takes 10 terms each and the overhead of simplifying. I found this website which lets you do symbolic manipulations on the web and the answer I got was 21952+15360*sqrt(2)+12544*sqrt(3)+8960*sqrt(6).

There is another interesting problem and that involves proving that Sum(comb(n,k)*D(k),k = 0..n) = n!. I tried using simple cases and it works out ok.