The 4 sticks in a complete bag of Pick-Up Sticks are all

Question

The 4 sticks in a complete bag of Pick-Up Sticks are all straight-line segments of negligible width, but each has a different length: 1 inch, 2 inches, 3 inches, and 4 inches, respectively. If Tommy picks a stick from each of 3 different complete bags of Pick-Up Sticks, what is the probability that Tommy CANNOT form a triangle from the 3 sticks?
(A) 11/32
(B) 13/32
(C) 15/32
(D) 17/32
(E) 19/32

A. 11/32 
B. 13/32 
C. 15/32 
D. 17/32 
E. 19/32

cyberjadugar · Answer

Hi, That the manhattan challenge problem for this week! One should hold off the answer till week end. :twisted:

Regards,

gnan · Answer

Here's my solution.

Total number of ways of picking sticks = 4*4*4 = 64 (As Tommy can pick any of the four sticks from each bag)

A Triangle can be formed only if the lengths of the sticks are any of the following pairs:
(1,1,1),(2,2,2),(3,3,3),(4,4,4) - 1 way
(2,2,1),(2,2,3),(3,3,1),(3,3,2),(3,3,4),(4,4,1),(4,4,2),(4,4,3) - 3 ways....Example (2,2,1) can be either (1,2,2) or (2,1,2) or (2,2,1)
(2,3,4) - 6 ways

Total ways of forming a triangle = 34
Probability of forming triangle = 34/64

So, Probability of NOT forming a triangle = 30/64 = 15/32
Ans:C

cyberjadugar · Answer

Hi,

My solution:
Please note  ()  - not possible
There are three bags with 4 sticks each:
(1, 2, 3, 4), (1, 2, 3, 4) & (1, 2, 3, 4), Now from each bag one stick has to be chosen.
Case 1: When all sticks are of same lenght (1, 1, 1), (2, 2, 2), (3, 3, 3) or (4, 4, 4)
4 possibilities.

Case 2: When two are of same lenght, and checking the property of trangles such that  |a-b|<c<a+b 
 (1, 1, 2)  (1, 2, 2)  (1, 1, 3)  (1, 3, 3)
 (1, 1, 4)  (1, 4, 4) (2, 2, 3) (2, 3, 3)
 (2, 2, 4)  (2, 4, 4) (3, 3, 4) (3, 4, 4)
8 possibilities;
each possible case can be drawn in different order: thus 8*3!/2!=24

Case 3: When all are different;  (1, 2, 3) (1, 2, 4) (1, 3, 4)  (2, 3, 4),
can be drawn in 3! ways = 6

Considering all above cases: 4+24+6 = 34 ways in which sticks can be drawn to form a triangle.
Total ways = 4C1*4C1*4C1=64
Number of ways when stick is drawn and no triangle is formed = 64-34=30
Probability = 30/64 = 15/32

Answer (C).

cyberjadugar · Answer

Hi,

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Regards,

The 4 sticks in a complete bag of Pick-Up Sticks are all

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