Hi! Anyone can help me with any/some/all of the following questions:
1. Determine the number of 4-element subsets {a, b, c, d} of {1, 2, 3, 4,..., 20} such that a + b + c + d is divisible by 3.
Pretty sure for this one I'll need to use some mod, not sure how though
2. There are 10! permutations s0s1...s9 of 0, 1,...,9. How many of them satisfy sk > k-2 for k = 0, 1,...9?
3. Let S = {1, 2, 3, 4, ..., 50}. Find the number of 3-element subsets {a,b,c} of S that satisfy the following: a + b + c is divisible by 3
4. Let P be a 30-sided polygon inscribed in a circle. Find the number of triangles whose vertices are the vertices of P such that any 2 vertices of each triangle are separated by at least 3 other vertices of P.
Thanks!!