If the total number of ways of selecting two numbers from the set `{1, 2, 3, ……….., 89, 90}` such that their sum is divisible by 3 is k, then `(k)/(500)` is

Answer

To solve the problem, we need to find the total number of ways to select two numbers from the set `{1, 2, 3, ..., 90}` such that their sum is divisible by 3. Let's break down the solution step by step.

### Step 1: Classify the Numbers by Remainders
When dividing numbers by 3, they can yield three possible remainders: 0, 1, or 2. We will classify the numbers from 1 to 90 based on these remainders.

- **Numbers divisible by 3 (remainder 0)**: These are 3, 6, 9, ..., 90. The sequence is an arithmetic sequence where:
- First term (a) = 3
- Common difference (d) = 3
- Last term (l) = 90
- Number of terms (n) can be calculated using the formula for the nth term of an arithmetic sequence:
\[ n = \frac{l - a}{d} + 1 = \frac{90 - 3}{3} + 1 = 30 \]

- **Numbers giving remainder 1**: These are 1, 4, 7, ..., 88. The sequence is:
- First term (a) = 1
- Common difference (d) = 3
- Last term (l) = 88
- Number of terms (n):
\[ n = \frac{88 - 1}{3} + 1 = 30 \]

- **Numbers giving remainder 2**: These are 2, 5, 8, ..., 89. The sequence is:
- First term (a) = 2
- Common difference (d) = 3
- Last term (l) = 89
- Number of terms (n):
\[ n = \frac{89 - 2}{3} + 1 = 30 \]

### Step 2: Count the Combinations
To find pairs of numbers whose sum is divisible by 3, we can have the following combinations:

1. **Both numbers are divisible by 3**: We can choose 2 numbers from the 30 numbers that are divisible by 3.
\[ \text{Ways} = \binom{30}{2} = \frac{30 \times 29}{2} = 435 \]

2. **One number gives remainder 1 and the other gives remainder 2**: We can choose 1 number from the 30 numbers that give remainder 1 and 1 number from the 30 numbers that give remainder 2.
\[ \text{Ways} = \binom{30}{1} \times \binom{30}{1} = 30 \times 30 = 900 \]

### Step 3: Total Ways
Now we can add the two cases together to get the total number of ways \( k \):
\[ k = 435 + 900 = 1335 \]

### Step 4: Calculate \( \frac{k}{500} \)
Finally, we need to find \( \frac{k}{500} \):
\[ \frac{k}{500} = \frac{1335}{500} = 2.67 \]

### Final Answer
Thus, the final answer is:
\[ \frac{k}{500} = 2.67 \]