ktrout2020 wrote: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
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
