71*. a) Banach's problem ${ }^{1}$. A person simultaneously bought two boxes of matches and put them in his pocket. After that, every time he needed to light a match, he randomly took one or the other box. After some time, upon emptying one of the boxes, the person discovered that it was empty. What is the probability that at this moment the second box still contained $k$ matches, if the number of matches in the unopened box was $n$?

b) Using the result from part a), find the value of the sum

$$
C_{2 n}^{n}+2 C_{2 n-1}^{n}+4 C_{2 n-2}^{n}+\ldots+2^{n} C_{n}^{n}
$$

Note. Another method for finding this sum is given in the solution to problem 55.