Consider all 3 element subsets of the set {1, 2, 3, ………. 300} then:

Options

A. A. Number of these subsets for which Sum of the three elements is multiple of 3 is `3xx""^(100)C_(3)+100^(3)`
B. B. Number of these subsets for which sum of the three elements is not a multiple of 3 is `3xx""^(10)C_(3)+100^(3)`
C. C. Number of these subsets for which sum of the three elements is even is `""^(151)C_(3)+149xx""^(150)C_(2)`
D. D. Numberof these subsets for which sum of the three elements is even is `""^(300)C_(2)-""^(150)C_(3)`

Answer

To solve the problem of finding the number of 3-element subsets of the set {1, 2, 3, ..., 300} such that the sum of the elements in each subset is a multiple of 3, we can follow these steps:

### Step 1: Define the Set
We have the set \( S = \{ 1, 2, 3, \ldots, 300 \} \).

### Step 2: Classify Elements by Remainders
When dividing by 3, the elements of the set can be classified into three categories based on their remainders:
- \( S_1 \): Elements that give a remainder of 1 when divided by 3 (i.e., \( 3n + 1 \))
- \( S_2 \): Elements that give a remainder of 2 when divided by 3 (i.e., \( 3n + 2 \))
- \( S_3 \): Elements that are multiples of 3 (i.e., \( 3n \))

### Step 3: Count the Elements in Each Subset
For the set \( S \):
- \( S_1 = \{ 1, 4, 7, \ldots, 298 \} \): This is an arithmetic sequence with the first term 1 and a common difference of 3. The number of terms can be calculated as follows:
\[ n_1 = \frac{298 - 1}{3} + 1 = 100 \]

- \( S_2 = \{ 2, 5, 8, \ldots, 299 \} \): Similarly,
\[ n_2 = \frac{299 - 2}{3} + 1 = 100 \]

- \( S_3 = \{ 3, 6, 9, \ldots, 300 \} \):
\[ n_3 = \frac{300 - 3}{3} + 1 = 100 \]

Thus, we have \( n_1 = n_2 = n_3 = 100 \).

### Step 4: Calculate the Valid Combinations
To form a 3-element subset whose sum is a multiple of 3, we can have the following cases:
1. All three elements from \( S_1 \): \( \binom{100}{3} \)
2. All three elements from \( S_2 \): \( \binom{100}{3} \)
3. All three elements from \( S_3 \): \( \binom{100}{3} \)
4. One element from each of \( S_1, S_2, S_3 \): \( \binom{100}{1} \times \binom{100}{1} \times \binom{100}{1} = 100^3 \)

### Step 5: Total Count of Valid Subsets
The total number of valid subsets is given by:
\[ \text{Total} = \binom{100}{3} + \binom{100}{3} + \binom{100}{3} + 100^3 = 3 \times \binom{100}{3} + 100^3 \]

### Step 6: Final Expression
Thus, the final expression for the number of 3-element subsets where the sum is a multiple of 3 is:
\[ \text{Total} = 3 \times \binom{100}{3} + 100^3 \]