If x, y and z are non-negative integers such that x < y < z,

X, Y, Z Is non negative integer, and x+y+z=11, and x<y<z
We can fix value of x and calculate distinct solutions
Case-1
X=0
Y can have 1,2,3,4,5 (Total 5 values) to reach x+y+z =11.
For each value of y z will have a fix value. So total 5 distinct solutions.
Case-2
x=1
Y can be 2,3,4, for each value, z will have a fix value ( 6,7, and 8)
Total 3 distinct solutions in this case.
Case -3
X=2
Y can be 3,4 for each value, z will have a fix value ( 5,6,)
Total 2 distinct solutions in this case.
Case -4
X=3
No value of y can satisfy x+y+z=11, and x<y<z.

Total
5+3+2=10
B.

Misinterpreted non-negative for positive.

I created this question to show that there can be times when the best (i.e., fastest) way to solve a counting question is by listing and counting

How do we know when it's not a bad idea to use listing and counting?
The answer choices will tell us (ALWAYS scan the answer choices before beginning any answer choices)
Here, the answer choices are reasonably small, so listing and counting shouldn't take long.
ASIDE: Yes, 78 (answer choice E) is pretty big. However, if you start listing possible outcomes and you eventually list more than 22 outcomes (answer choice D), you can stop because the answer must be E.

Okay, let's list possible outcomes in a systematic way.
We'll list outcomes as follows: x, y, z

Since x is the smallest value.
Let's list the outcomes in which x = 0. We get:
0, 1, 10
0, 2, 9
0, 3, 8
0, 4, 7
0, 5, 6

Now, the outcomes in which x = 1. We get:
1, 2, 8
1, 3, 7
1, 4, 6

Now, the outcomes in which x = 2. We get:
2, 3, 6
2, 4, 5

Now, the outcomes in which x = 3. We get:
3, 4...hmmm, this won't work.

So, we're done listing!
Count the outcomes to get a total of 10

Answer: B

Cheers,
Brent

0≤x<y<z
x+y+z=11
x=0, y=1, z=11-1=10: average (10+1)=11/2=5.5 so y={1 to 5}={1,2,3,4,5}=5
x=1, y=2, z=11-3=8: average (8+2)=10/2=5 so y={2 to 4}={2,3,4}=3
x=2, y=3, z=11-5=6: average (6+3)=9/2=4.5 so y={3 to 4}={3,4}=2
x=3, y=4, z=11-7=4: invalid

total solutions: 5+3+2=10

Answer (B)

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