Question

If \(x+y+z=17\) and x, y, z are non-negative integers, then find the number of integral solutions is

• A. 136
• B. 171
• C. 90
• D. 130

Answer 1

The Correct Option is B

To solve this problem, we need to find the number of non-negative integer solutions for the equation \(x+y+z=17\).

This type of problem can be solved using the stars and bars method in combinatorics. The idea is to find the number of ways to distribute n identical items (in this case, the number 17) into r distinct groups (here, the variables \(x\), \(y\), \(z\)) where each group can hold zero or more items.

The formula for finding the number of non-negative integer solutions to the equation \(x_1 + x_2 + \cdots + x_r = n\) is given by:

\(\binom{n + r - 1}{r - 1}\)

For our specific equation \(x+y+z=17\), here \(n=17\) and \(r=3\), so we substitute these into the formula:

\(\binom{17 + 3 - 1}{3 - 1} = \binom{19}{2}\)

Now, calculate \(\binom{19}{2}\):

\(\binom{19}{2} = \frac{19 \times 18}{2 \times 1} = \frac{342}{2} = 171\)

Thus, the number of non-negative integer solutions for the equation \(x+y+z=17\) is 171.

Among the given options, 171 is the correct answer.