Problems

Consider the \(4!\) possible permutations of the numbers \(1,2,3,4\). Which of those permutations keep the expression \(x_1x_2+x_3x_4\) the same?

Show that if \(1+3+5+7+...+97+99=50^2\), then \(1+3+5+7+...+97+99+101=51^2\). Don’t forget that \((a+b)^2=a^2+2ab+b^2\).

Prove that for all positive integers \(n\) there exists a partition of the set of positive integers \(k\le2^{n+1}\) into sets \(A\) and \(B\) such that \[\sum_{x\in A}x^i=\sum_{x\in B}x^i\] for all integers \(0\le i\le n\).

In Pascal’s triangle, what are the numbers in the diagonal next to the diagonal of ones?

In Pascal’s triangle, what is the sum of the entries in each row?

Oliver throws a fair coin three times. What are his chances of getting three heads, two heads and one tail, one head and two tails, or three tails?

In Pascal’s triangle, what numbers appear in the diagonal next to the positive integers?

Five friends get together and want to take a photo. They all agree that two of them should take a photo of the other three. How many ways can you choose the three people to be in the picture?

In Pascal’s triangle, what’s the sum of the numbers in each row when you put a minus sign in front of every other number?

What is \(11^2\), \(11^3\) and \(11^4\), and what do these numbers have to do with Pascal’s triangle? What goes wrong with \(11^5\)?