Fifteen dots are evenly spaced on the circumference of a circle. How many combinations of three dots can we pick from

BTGmoderatorDC wrote: ↑

Fri Mar 13, 2020 4:32 pm

Fifteen dots are evenly spaced on the circumference of a circle. How many combinations of three dots can we pick from these 15 that do not form an equilateral triangle?

A. 160
B. 450
C. 910
D. 1360
E. 2640


OA B

Source: Magoosh

KEY CONCEPT
If we connect ANY 3 dots, we'll get a unique triangle.
Since the order in which we select the dots does not matter, we can use COMBINATIONS.
We can select 3 dots from 15 dots in 15C3 ways
15C3 = (15)(14)(12)/(3)(2)(1) = 455

Now SOME of these 455 triangles will be equilateral triangles, so we must subtract from 455 the number of those triangles that are equilateral triangles.

IMPORTANT: Since the correct answer must be less than 455, we can ELIMINATE answer choices C, D and E

At this point, we COULD determine the number of equilateral triangles that are included among the 455 triangles we've counted (the above posters have already done so).
However, we could use the answer choices to our advantage.

Answer choice A (160) suggests that there are 295 equilateral triangles among the 455 triangles we've counted (since 455 - 295 = 160)
Answer choice B (450) suggests that there are 5 equilateral triangles among the 455 triangles we've counted (since 455 - 5 = 450)

If answer A is correct, then more than half of the 455 triangles are equilateral triangles. This doesn't seem right since MOST selections of 3 points will NOT yield an equilateral triangle.
So, ELIMINATE A

Answer: B

Cheers,
Brent

BTGmoderatorDC wrote: ↑

Fri Mar 13, 2020 4:32 pm

Fifteen dots are evenly spaced on the circumference of a circle. How many combinations of three dots can we pick from these 15 that do not form an equilateral triangle?

A. 160
B. 450
C. 910
D. 1360
E. 2640


OA B

Source: Magoosh

The total number of triangles that can be formed (regardless of whether they are equilateral) is 15C3 = (15 x 14 x 13)/(3 x 2) = 5 x 7 x 13 = 455. Now let’s determine the number of triangles formed that are equilateral triangles.

Let A, B, C, D, E, F, G, H, I, J, K, L, M, N, and O be the 15 points that are evenly spaced on the circumference of the circle. Every pair of consecutive points (for example, A and B, G and H, etc.) form a 360/15 = 24-degree arc. In order for 3 points to form an equilateral triangle, they must evenly spaced among themselves on the circumference of the circle also. That is, each pair of the three points have to be 360/3 = 120 degrees apart. Since 120/24 = 5, the points should be 5 spaces apart from one another. For example, if A is one of the vertices of the equilateral triangle, then the next one should be F and the last one should be K. In other words, triangle AFK is an equilateral triangle. Using the same analogy, triangles BGL, CHM, DIN, and EJO are equilateral triangles and only these 5 (including triangle AFK) are equilateral triangles.

Since 455 triangles can be formed and 5 of these are equilateral, then the number of triangles that are not equilateral is 455 - 5 = 450.

Answer: B