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 of selecting two numbers from the set `{1, 2, 3, …, 89, 90}` such that their sum is divisible by 3, we can break down the solution into a series of steps.

### Step-by-Step Solution:

1. **Identify the Set**: The set consists of the numbers from 1 to 90.

2. **Classify Numbers by Remainders**: When divided by 3, numbers can give remainders of 0, 1, or 2. We need to count how many numbers fall into each category:
- **Remainder 0**: These are multiples of 3. The multiples of 3 from 1 to 90 are: 3, 6, 9, ..., 90. The total count is \( \frac{90}{3} = 30 \).
- **Remainder 1**: These numbers are of the form 1, 4, 7, ..., 88. The total count is also \( \frac{90 - 1}{3} + 1 = 30 \).
- **Remainder 2**: These numbers are of the form 2, 5, 8, ..., 89. The total count is \( \frac{90 - 2}{3} + 1 = 30 \).

3. **Calculate Ways to Select Pairs**:
- **Case 1**: Both numbers give a remainder of 0 (i.e., both are multiples of 3). The number of ways to choose 2 from 30 is given by the combination formula \( \binom{n}{r} \):
\[ \text{Ways} = \binom{30}{2} = \frac{30 \times 29}{2} = 435 \]

- **Case 2**: One number gives a remainder of 1 and the other gives a remainder of 2. The number of ways to choose one from each group is:
\[ \text{Ways} = \binom{30}{1} \times \binom{30}{1} = 30 \times 30 = 900 \]

4. **Total Ways**: Add the number of ways from both cases:
\[ k = 435 + 900 = 1335 \]

5. **Calculate \( \frac{k}{500} \)**:
\[ \frac{k}{500} = \frac{1335}{500} = 2.67 \]

### Final Answer:
\[ \frac{k}{500} = 2.67 \]