Question

determine all combinations of lengths that vould be usef to for a traingle 4,6,12,16 and also another lengths of 8,10,14,17

Answer 1

`\lbrace 6,\, 12,\, 16 \rbrace`.

`\lbrace 8,\, 10,\, 14 \rbrace`,

`\lbrace 8,\, 10,\, 17\rbrace`,

`\lbrace 8,\, 14,\, 17 \rbrace`, and

`\lbrace 10,\, 14,\, 17 \rbrace`.

Explanation:

In a triangle, the sum of the lengths of any two sides must be strictly greater than the length of the other side. In other words, if
`a`,
`b`, and
`c` (where
`a,\, b,\, c > 0`) denote the lengths of the three sides of a triangle, then
`a + b > c`,
`a + c > b`, and
`b + c > a`.

Additionally, if
`a,\, b,\, c > 0`,
`a + b > c`,
`a + c > b`, and
`b + c > a`, then three line segments of lengths
`\text{$a$, $b$, and $c$}`, respectively, would form a triangle.

Number of ways to select three numbers out of a set of four distinct numbers:

`\begin{aligned} \left(\begin{matrix}4 \\ 3\end{matrix}\right) &= (4!)/(3! \, (4 - 3)!) \\ &= (4 * 3 * 2 * 1)/((3 * 2 * 1) * (1)) \\ &= 4\end{aligned}`.

In other words, there are four ways to select three numbers out of a set of four distinct numbers.

For
`\lbrace 4,\, 6,\, 12,\, 16\rbrace`, the four ways to select the three numbers are:

• 
  `\lbrace 4,\, 6,\, 12\rbrace`,
• 
  `\lbrace 4,\, 6,\, 16\rbrace`,
• 
  `\lbrace 4,\, 12,\, 16\rbrace`, and
• 
  `\lbrace 6,\, 12,\, 16\rbrace`.

Among these choices,
`\lbrace 6,\, 12,\, 16\rbrace` satisfies the requirements:
`6 + 12 > 16`,
`16 + 6 > 12`, and
`12 + 16 > 6`. Thus, three line segments of lengths
`\lbrace 6,\, 12,\, 16\rbrace\!` respectively would form a triangle.

The other three combinations do not satisfy the requirements. For example,
`\lbrace 4,\, 6,\, 12\rbrace` does not satisfy the requirements because
`4 + 6 < 12`. The combination
`\lbrace 4,\, 12,\, 16\rbrace` does not satisfy the requirement for a slightly different reason. Indeed
`4 + 12 = 16`. However, the inequality in the requirement needs to be a strict inequality (i.e., strictly greater than "
`>`" rather than greater than or equal to "
`\ge`".)

Thus,
`\lbrace 6,\, 12,\, 16\rbrace` would be the only acceptable selection among the four.

Similarly, for
`\lbrace 8,\, 10,\, 14,\, 17 \rbrace`, the choices are:

• 
  `\lbrace 8,\, 10,\, 14 \rbrace`,
• 
  `\lbrace 8,\, 14,\, 17 \rbrace`,
• 
  `\lbrace 8,\, 10,\, 17 \rbrace`, and
• 
  `\lbrace 10,\, 14,\, 17 \rbrace`.

All four combinations satisfy the requirements.