Permutations and Combinations Hard Level

How many different 4-letter arrangements can be made from the letters of BANANA?

5 points are marked on one line and 6 points on a parallel line. How many triangles have their vertices among these 11 points?

5 points lie on one line and 6 on a parallel line. How many quadrilaterals have their vertices among them?

In how many ways can 6 identical balls be placed in 3 identical boxes (boxes may be empty)?

How many functions from a set of 4 elements onto a set of 2 elements are there?

A cricket XI must contain exactly 4 bowlers, 6 batsmen and 1 wicket-keeper, chosen from 6 bowlers, 8 batsmen and 2 keepers. In how many ways?

The street grid below is 4 blocks by 4 blocks. Routes go from A to B, moving only right or up.

How many shortest routes from A to B pass through X?

Three points are marked on each side of a triangle, as shown (the corners are not among the marked points).

How many triangles have all three vertices among the nine marked points?

Three points are marked on each side of a triangle, as shown (the corners are not among the marked points).

How many different straight lines pass through at least two of the nine marked points?

How many positive integers below 1,000 contain no digit 0?