A box contains 5 red marbles, 8 green marbles, 11 blue marbles, and 15 yellow marbles. We draw marbles uniformly at random without replacement from the box. What is the minimum number of marbles to be drawn to ensure that out of the marbles drawn, at least 7 are of the same colour?

The maximum area of a rectangle inscribed in the unit circle (i.e., all the vertices of the rectangle are on the circle) in a plane is:

Let M be an n m real matrix. Consider the following:
* Let k1 be the smallest number such that M can be factorized as AB, where A is an n x k1 and B is a k1 x m matrix.
*Let k2 be the smallest number such that M = Pk2 i=1 uivi, where each u_i is an Nx1 matrix and each v_i is an 1 x m matrix.
* Let k3 be the column-rank of M.
Which of the following statements is true?

What is the probability that at least two out of four people have their birthdays in the same month, assuming their birthdays are uniformly distributed over the twelve months?

A matching in a graph is a set of edges such that no two edges in the set share a common vertex. Let G be a graph on n vertices in which there is a subset M of m edges which is a matching. Consider a random process where each vertex in the graph is independently selected with probability 0 < p < 1 and let B be the set of vertices so obtained. What is the probability that there exists at least one edge from the matching M with both end points in the set B?

Let d be the number of positive square integers (that is, it is a square of some integer) that are factors of 20⁵ X 21⁵. Which of the following is true about d?

Fix n > 6. Consider the set C of binary strings x₁x₂... x_n of length n such that the bits satisfy the following set of equalities, all modulo 2: x_i + x_i+1 + x_i+2 = 0 for all 1 < i < n - 2, xn-1 + xn + x1 = 0, and xn + x1 + x2 = 0. What is the size of the set C?

Lavanya and Ketak each flip a fair coin n times. What is the probability that Lavanya sees more heads than Ketak? In the following, the binomial coefficient (n/k) counts the number of k-element subsets of an n-element set.

Find the following sum.

How many numbers in the range f0; 1; : : : ; 1365g have exactly four 1’s in their binary representation? (Hint: 136510 is 101010101012, that is, 1365 = 2¹⁰ + 2⁸ + 2⁶ + 2⁴ + 2² + 2⁰🙂 In the following, the binomial coefficient (n/k) counts the number of k-element subsetsof an n-element set.