Question

Given six line segments of length $2,3,4,5,6,7$ units, the number of triangles that can be formed by these segments is

• A. ${}^6{C}_3-7$
• B. ${}^6{C}_3-6$
• C. ${}^5{C}_3-5$
• D. ${}^5{C}_3-4$

Answer 1

The correct option is A ${}^6{C}_3-7$

We have the numbers $2,3,4,5,6,7$.

Sum of any two sides of a triangle must be greater than the third side.

Equivalently, the sum of the two shortest sides must be greater than the third side.

We are making a triangle. The shortest stick we use can be any of $2,3,4,5$.

We find for each choice of shortest stick, how many bad choices there are for the other two sticks. (Bad choice = No triangle.)

Shortest stick is 2: If next shortest stick is $3$, then $5,6,$ and $7$ for third are bad.

If next shortest after $2$ is $4$, then $5$ and $6$ for third are bad.

If next shortest after $2$ is $5$, then $7$ for third is bad.

Total bad here: $3+2+1=6$.

Shortest stick is 3: If next shortest stick is $4$, then $7$ for third is bad

Shortest is $4$ or $5$: No bads.

No we have total $6+1$ bad combinations.

We have total ${}^6{C}_3$

After removing bads we get ${}^6{C}_3-7$ combinations.