Bunuel wrote:

There are six marked points on the circle above. How many different lines can be drawn that contain two of the marked points?

(A) 5

(B) 6

(C) 12

(D) 15

(E) 30

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While I agree that an approach using combinations probably is efficient for most, a very easy pattern exists here.

The pattern-finding comes in handy when under pressure and, ahem, temporarily grouchy about combinatorics.

The method was not a time-killer. 40 seconds.

Label the points to track on where you started.

Draw, and count, all possible lines from A. There will be

AB, AC, AD, AE, and AF = 5

Draw and count all possible lines from B that do not replicate the line already drawn, AB. There will be

BC, BD, BE, and BF = 4

Draw and count all possible lines from C without replication of lines already drawn. There will be

CD, CE, and CF = 3. Stop.

The pattern is clear: the number of distinct lines that can be drawn from successive points, except F, decreases by 1 for each point.

(You can't draw anything new from F. It has been a point on one of every other letter's lines. Lines drawn from F = 0.)

Number of lines that contain exactly two points = (5 + 4 + 3 + 2 + 1 + 0) = 15

Answer (D)