Binomial Coefficients

Question

Using Combinatorial How can i Calculate

(1) The Sum $\displaystyle \bf{\sum_{i=0}^{n}\binom{10}{i}.\binom{20}{m-i} = Maximum}$ . Then value of $\bf{m}$ is

(2) The sum of $\displaystyle \bf{\binom{30}{0}.\binom{30}{10}-\binom{30}{1}.\binom{30}{11}+........................+\binom{30}{20}.\binom{30}{30}=}$

I have solved These Question Using expansion of $\bf{(1+x)^n}$ and $\bf{(1-x)^n}$.

But Someone told me that we can easily solved in less no. of steps using Combinatorial Method.

Can anyone explain me how can i solved it.

Thanks

Answer 1

For (1): How many ways can you pick $m$ objects from 30 objects? You can pick $i$ from the first $10$ objects and $m-i$ from the next 20...

For the second one:

$$\binom{30}{0}\binom{30}{20}-\binom{30}{1}\binom{30}{19}+........................+\binom{30}{20}\binom{30}{0}$$

Think about it this way: You have 30 blue objects and 30 red objects.

Then the sum represents the number of way of choosing 20 objects so that the number of red and blue is even - the number of way of choosing 20 objects so that the number of red and blue is odd.

You should be able for each of the two parts to figure the right formula and get an easy closed form.