# Thread: Some basic combinatorics problems

1. ## Some basic combinatorics problems

(I'm not sure if this is in the right forum... since there is no forum that says "combinatorics")

I've just started combinatorics (basic) and just some beginner questions I can't do, if anyone could help with them it would be very much appreciated :lol: [Note: if it's possible please don't leave out any necessary steps or procedures I'm a beginner at these and pretty n00b haha]

1. Prove $\left(^n_0\right)+\left(^n_1\right)+\left(^n_2\right)+...+\left(^n_n\right) = 2^n$

2. Which are there more of among the natural numbers between $1$ and $10^6$: Numbers that can be represented as a sum of a perfect square and a (positive) perfect cube or numbers that can not be?

3. Two of the squares of a $7 \times 7$ checkerboard are painted yellow and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible?

2. For the first, consider Newton's binomial expansion:

$\displaystyle (a+b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n}b^{n-k}$

What happens if we let a=b=1?

3. ## Question 2

Up to 10^6, there are 10^3 squares. Similarly, there are 10^2 cubes. Optimistically assuming there are no duplicates, there are 10^5 = 10^3 * 10^2 possible pairs, leaving 9 x 10^5 unaccounted for numbers in 10^6.