Duong noticed that _{$\large{_{1}C_{0} + _{1}C_{1} = 2^{1}}$}. He tried _{$\large{_{2}C_{0} + _{2}C_{1} + _{2}C_{2} = 2^{2}}$} and found that it worked also.

1. Does _{$\large{_{3}C_{0} + _{3}C_{1} + _{3}C_{2} + _{3}C_{3} = 2^{3}}$}?

   Hint (a):

   Use your calculator to test the expression.

   Answer (a):

   Yes, it also works.

2. Does _{$\large{_{4}C_{0} + _{4}C_{1} + _{4}C_{2} + _{4}C_{3} + _{4}C_{4} = 2^{4}}$}?

   Hint (b):

   See part (a).

3. Explain why _{$\large{_{n}C_{0} + _{n}C_{1} + _{n}C_{2} + …+ _{n}C_{n} = 2^{n}}$} .

   Hint (c):

   Any reasonable explanation is acceptable.