Question

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?

Answer 1

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

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 =
`^(5+3-1)C_(3-1)=^7C_2=21\:ways`

⇒ There are 21 positive solutions

b).

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

Number of ways =
`^(8+3-1)C_(3-1)=^(10)C_2=45\:ways`

⇒ There are 45 negative solutions.

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