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

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 [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex] (where [tex]a,\, b,\, c > 0[/tex]) denote the lengths of the three sides of a triangle, then [tex]a + b > c[/tex], [tex]a + c > b[/tex], and [tex]b + c > a[/tex].

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

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

[tex]\begin{aligned} \left(\begin{matrix}4 \\ 3\end{matrix}\right) &= \frac{4!}{3! \, (4 - 3)!} \\ &= \frac{4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1) \times (1)} \\ &= 4\end{aligned}[/tex].

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

For [tex]\lbrace 4,\, 6,\, 12,\, 16\rbrace[/tex], the four ways to select the three numbers are:

[tex]\lbrace 4,\, 6,\, 12\rbrace[/tex],
[tex]\lbrace 4,\, 6,\, 16\rbrace[/tex],
[tex]\lbrace 4,\, 12,\, 16\rbrace[/tex], and
[tex]\lbrace 6,\, 12,\, 16\rbrace[/tex].

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

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

Thus, [tex]\lbrace 6,\, 12,\, 16\rbrace[/tex] would be the only acceptable selection among the four.

Similarly, for [tex]\lbrace 8,\, 10,\, 14,\, 17 \rbrace[/tex], the choices are:

[tex]\lbrace 8,\, 10,\, 14 \rbrace[/tex],
[tex]\lbrace 8,\, 14,\, 17 \rbrace[/tex],
[tex]\lbrace 8,\, 10,\, 17 \rbrace[/tex], and
[tex]\lbrace 10,\, 14,\, 17 \rbrace[/tex].

All four combinations satisfy the requirements.