If `x+y le4` , then the how many non- zero positive integar ordered pair (x,y)?

Options

A. 4
B. 5
C. 6
D. 8

Explanation

To solve the problem of finding the number of non-zero positive integer ordered pairs \((x, y)\) such that \(x + y \leq 4\), we can follow these steps:

### Step 1: Understand the constraints
We know that \(x\) and \(y\) are non-zero positive integers. Therefore, the smallest value for both \(x\) and \(y\) is 1.

### Step 2: Set up the inequality
The inequality we need to satisfy is:
\[ x + y \leq 4 \]
Given that both \(x\) and \(y\) must be at least 1, we can rewrite the inequality in terms of \(x\) and \(y\):
\[ 1 \leq x \leq 3 \quad \text{and} \quad 1 \leq y \leq 3 \]

### Step 3: Find possible values for \(x\) and \(y\)
We can find the ordered pairs \((x, y)\) by considering the possible values for \(x\) and calculating the corresponding values for \(y\) such that the sum does not exceed 4.

1. If \(x = 1\):
- \(y\) can be \(1, 2, 3\) (since \(1 + y \leq 4\))
- Ordered pairs: \((1, 1), (1, 2), (1, 3)\) → 3 pairs

2. If \(x = 2\):
- \(y\) can be \(1, 2\) (since \(2 + y \leq 4\))
- Ordered pairs: \((2, 1), (2, 2)\) → 2 pairs

3. If \(x = 3\):
- \(y\) can be \(1\) (since \(3 + y \leq 4\))
- Ordered pair: \((3, 1)\) → 1 pair

### Step 4: Count the total number of pairs
Now, we can add the number of pairs from each case:
- From \(x = 1\): 3 pairs
- From \(x = 2\): 2 pairs
- From \(x = 3\): 1 pair

Total number of ordered pairs:
\[ 3 + 2 + 1 = 6 \]

### Final Answer
The total number of non-zero positive integer ordered pairs \((x, y)\) such that \(x + y \leq 4\) is **6**.

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