150 young adults were asked how they commuted to work. 125 said they take the underground and 93 said they cycle. Of all the people interviewed, 72 said they both cycle and take the underground. How many people do not cycle and do not take the underground?
Show that there are no more than 266 prime numbers less than or equal to 1000.
200 people were asked if they drank one of the following beverages regularly: tea, coffee and beer. 185 people said they drank at least one of these beverages. Funnily, for every choice of a pair of beverages, 122 people said they drank at least one beverage out of the pair. The even stranger fact was that for each choice of a beverage, exactly 73 people admitted to drinking it.
How many people drink all three beverages?
Each student chooses at least one from the \(n\) different modules offered at a university. Let us number these modules as \(1,2,3,\dots,n\). For each natural number \(1\leq k\leq n\), we denote the number of students choosing the modules \(i_1,\dots,i_k\) by \(S(i_1,\dots,i_k)\). Give a formula for the number of students in terms of the numbers \(S(i_1,\dots,i_k)\).
As an example, if \(n = 5\), \(k=3\) and we look at \(i_1 = 4,i_2 = 2, i_3 =1\), then \(S(i_1,i_2,i_3) = S(4,2,1)\) is the number of students picking the modules \(1,2,4\).
There are \(n\) seats on a plane and each of the \(n\) passenger sat in the wrong seat. What is the total number of ways this could happen?
Let \(n\geq 2\) be a integer. Fix \(2n\) points in space and select any \(n^2+1\) segments between these points. Show that these segments must form at least \(n\) triangles.