A triangle has two sides with lengths 7 and 12 which of the folowing lengths could represent the third side

If s₃ is the length of the 3rd side, the possible values are in the interval.

5 < s₃ < 19

Step-by-step explanation:

For a triangle, we know that the sum of any two sides is always larger than the other side.

So if a triangle has 3 sides:

s₁, s₂, and s₃, we have:

s₁ + s₂ > s₃

s₁ + s₃ > s₂

s₂ + s₃ > s₁

In this case, we know that two sides are:

s₁ = 7

s₂ = 12

And we want to find the possible values of s₃.

Then if we use the above inequalities, we get:

7 + 12 > s₃

7 + s₃ > 12

12 + s₃ > 7

With the first one we get:

19 > s₃

Now we can rewrite the other two as:

s₃ > 12 - 7 = 5

s₃ > 7 - 12 = -5

The first one is more restrictive than the second:

s₃ > 5 > -5

So we can use only the first one.

Then the two inequalities are

s₃ > 5

s₃ < 19

Then the range is:

5 < s₃ < 19

This means that any value between 5 and 19 can be a possible length of the third side (5 and 19 are not possible lengths)

a) There is a 1.82% probability of selecting a jury of all students.

b) There is a 0.02% probability of selecting a jury of all faculty.

c) There is a 36.4% probability of selecting a jury of three students and three faculty.

Step-by-step explanation:

6 individuals are going to be selected. There are no replacements.So:

(a) What is the probability of selecting a jury of all students?

Initially, there are 9 students and 15 people in the pool.

So, the probability that the first person selected to the jury is a student is .

Now, there are 8 students and 14 people in the pool. So, the probability that the second person selected to the pool is a student is .

We do this for the other four positions of the jury, and we end up with the following probability.

There is a 1.82% probability of selecting a jury of all students.

(b) What is the probability of selecting a jury of all faculty?

The same logic as a) above.

Initially, there are 6 faculty and 15 people in the pool.

So, the probability that the first person selected to the jury is a faculty is .

Now, there are 5 faculty and 14 people in the pool. So, the probability that the second person selected to the pool is a faculty is .

We do this for the other four positions of the jury, and we end up with the following probability.

There is a 0.02% probability of selecting a jury of all faculty.

(c) What is the probability of selecting a jury of three students and three faculty?

I am going to initially find the probability of finding the following sequence:

J-J-J-F-F-F.

Initially, there are 9 students, 6 faculty, that is 15 people in the pool.

So, the probability that the first person selected to the jury is a student is .

Now, there are 8 students and 14 people in the pool. So, the probability that the second person selected to the pool is a student is .

Now, there are 7 students and 13 people in the pool. So, the probability that the third person selected to the pool is a student is .

Now, there are 6 faculty and 12 people in the pool. So, the probability that the third person selected to the pool is a faculty is .

Now, there are 5 faculty and 11 people in the pool. So, the probability that the third person selected to the pool is a faculty is .

Now, there are 4 faculty and 10 people in the pool. So, the probability that the third person selected to the pool is a faculty is .

So, the probability of getting just this sequence is

However, any combination with 3 students and 3 faculty is valid, no matter the order. So, we need to multiply P by the combination of 6, with 3 by 3. So

There is a 36.4% probability of selecting a jury of three students and three faculty.