Clever counting » Combinations

Below you can find two examples of use of combinations.

Example 1

In a plane lie the points A to J, with no more than two points on a straight line.
How many different triangles can you make when you choose three of these points as vertices of a triangle?

Answer:
For the first vertex, you have 10 possibilities, for the second 9 and for the third 8.
However, the order of these vertices does not matter.
BDJ, BJD, DBJ, DJB, JBD and JDB are all the same triangle.
So the number of triangles is 3! = 6 times smaller.
The total number of possible triangles is not 10 · 9 · 8 but:
10 · 9 · 83! = 10 · 9 · 83 · 2 · 1 = 10 nCr 3 = 120


Example 2

In a vase are 8 red and 7 white marbles. You take 5 marbles from the vase.
In how many ways is it possible to take 3 red marbles?

Answer:
You take 3 of the 8 red marbles.
This can be done in bracket open8
3
bracket close = 8 · 7 · 63 · 2 · 1 = 8 nCr 3 = 56 ways.

You take 2 of the 7 white marbles.
This can be done in bracket open7
2
bracket close = 7 · 62 · 1 = 7 nCr 2 = 21 ways.

Therefore, there are 56 × 21 = 1176 ways to get 5 marbles of which 3 are red.