GMATPrepNow wrote:
If 3 different integers are randomly selected from the integers from 1 to 12 inclusive, what is the probability that a triangle can be constructed so that its 3 sides are the lengths of the 3 selected numbers?
A) 3/8
B) 7/18
C) 19/44
D) 39/88
E) 11/24
A student asked me to calculate the numerator for this question. So, here we go.....
ASIDE: This solution is beyond the scope of the GMAT. My intention with this question was to demonstrate the importance of calculating the denominator first.
Let's be systematic and arrange the lengths in
descending orderKEY CONCEPT: The longest side must be less than the sum of the other two sidesTriangle lengths with 12 as the longest side12, 11, 10
12, 11, 9
12, 11, 8
12, 11, 7
12, 11, 6
12, 11, 5
12, 11, 4
12, 11, 3
12, 11, 2
Total outcomes in the form 12, 11, _ = 9
12, 10, 9
12, 10, 8
12, 10, 7
12, 10, 6
12, 10, 5
12, 10, 4
12, 10, 3
Total outcomes in the form 12, 10, _ = 7
12, 9, 8
12, 9, 7
12, 9, 6
12, 9, 5
12, 9, 4
Total outcomes in the form 12, 9, _ = 5
12, 8, 7
12, 8, 6
12, 8, 5
Total outcomes in the form 12, 8, _ = 3
12, 7, 6
Total outcomes in the form 12, 7, _ = 1
So, the total number of outcomes with 12 as the longest side = 9 + 7 + 5 + 3 + 1= 25Triangle lengths with 11 as the longest side11, 10, 9
11, 10, 8
11, 10, 7
11, 10, 6
11, 10, 5
11, 10, 4
11, 10, 3
11, 10, 2
Total outcomes in the form 11, 10, _ = 8
11, 9, 8
11, 9, 7
11, 9, 6
11, 9, 5
11, 9, 4
11, 9, 3
Total outcomes in the form 11, 9, _ = 6
11, 8, 7
11, 8, 6
11, 8, 5
11, 8, 4
Total outcomes in the form 11, 8, _ = 4
11, 7, 6
11, 7, 5
Total outcomes in the form 11, 7, _ = 2
Total number of outcomes with 11 as the longest side = 8 + 6 + 4 + 2= 20Let's do one more round!
Triangle lengths with 10 as the longest side10, 9, 8
10, 9, 7
10, 9, 6
10, 9, 5
10, 9, 4
10, 9, 3
10, 9, 2
Total outcomes in the form 10, 9, _ = 7
Total outcomes in the form 10, 8, _ = 5
Total outcomes in the form 10, 7, _ = 3
Total outcomes in the form 10, 6, _ = 1
Total number of outcomes with 10 as the longest side = 7 + 5 + 3 + 1 = 16-------------------------------------------------
Let's summarize what we have so far:
Total number of outcomes with 12 as the longest side = 9 + 7 + 5 + 3 + 1= 25Total number of outcomes with 11 as the longest side = 8 + 6 + 4 + 2 = 20Total number of outcomes with 10 as the longest side = 7 + 5 + 3 + 1 = 16See the patterns of ODDS and EVENS?
Keep going to get:
The total number of outcomes with 9 as the longest side = 6 + 4 + 2 =
12 The total number of outcomes with 8 as the longest side = 5 + 3 + 1 =
9 The total number of outcomes with 7 as the longest side = 4 + 2 =
6 The total number of outcomes with 6 as the longest side = 3 + 1 =
4 The total number of outcomes with 5 as the longest side = 2 =
2 The total number of outcomes with 4 as the longest side =
1 At this point we're done.
So, the total number of triangles possible =
25 + 20 + 16 + 12 + 9 + 6 + 4 + 2 + 1=
95Since we already learned (from earlier posts) that the denominator =
220 So, P(creating a triangle) =
95/
220 = 19/44
Answer: C
Cheers,
Brent