combinatorics & binomial theorem

in question no 4 the answer is 217 as 1296-r shud be divisible by 2 and r shud be divisible by 3...
therefore r shud be divisible by 6....therefore 1296/6 terms...
this cums out to be 216 but we did not count for r to be 0....
so 217 terms....

1) should be 66*10!
3) use the fact that ⁿC_r = ⁿC_n-r to sum up the series

i think this can be done.
it will be C(2n,n).
try multiplying (1+x)ⁿ and (x+1)ⁿ
and compare the coefficients of the required terms...

@h4hemang,what else did i do? :P

post #4

Q1)
how to do it???
i am getting a different answer./.
something to do with partitions.

2) better do it like this,
c₀² + c₁².....+c_n²
= c₀*c_n + c₁*c_{n-1 ....}
= number of ways of selecting n objects out of 2n
= 2n_Cn