How many solutions are there for the equation x + y + z = 8 for which a) x, y, and z are all positive? b) x, y, and z are all non-negative?

a). There are 21 positive solutions    and    b). There are 45 solutions.

Step-by-step explanation:

Given:

Equation, x + y + x = 8

To find: a).  x, y, and z are all positive

             b). x, y, and z are all non-negative

Here, Finding the number of solutions is equivalent to finding the number of ways to distribute 8 objects among 3 places.

a).

First let us given each place 1 object each.

Now, we find the number of ways to distribute 5 objects among three places.

Number of ways =

⇒ There are 21 positive solutions

b).

Here, we find the number of ways to distribute 8 objects among three places.

Number of ways =

⇒ There are 45 negative solutions.

Therefore, a). There are 21 positive solutions    and    b). There are 45 solutions.

introduction, methodology, results, discussion

step-by-step explanation:

an observational study is one which consists of actual observation to record data and then analyse the data.

there are a number of factors that should be analyzed before starting an observational study.

the main four factors are:

i) introduction:   this should be analysed to check whether the purpose of the study is correctly established and the data recorded are appropriate for the study

methodology:   the method of collecting the data, such as whether randomly selected, and whether sample size matches with the population to be studied, whether questions asked are easy to answer by common men, etc.

results: then results such as summary of data with comments should be studied thoroughly

discussion:   the last is the discussion, this can be either by management people or persons who are all interested , etc.

true !

step-by-step explanation: