If k = `(k_1, k_2, k_3, ...., k_n) in I` and `k_1k_2k_3 + k_2k_3k_4 + k_3k_4k_5 + .... + k_(n-2) k_(n-1) k_n = 0`
Then minimum how many entities i.e.`k_i (i = 1,2,3, .....) `must be zero? If there are total 12 terms in the above expression :

Options

A. 3
B. 4
C. 6
D. `"^nC_3`

Answer

To solve the problem, we need to analyze the expression given:

\[ k_1k_2k_3 + k_2k_3k_4 + k_3k_4k_5 + \ldots + k_{n-2}k_{n-1}k_n = 0 \]

We have a total of 12 terms in this expression. Each term is a product of three consecutive elements from the sequence \( k_1, k_2, \ldots, k_n \).

### Step 1: Understanding the terms
Each term in the expression is formed by multiplying three consecutive \( k \) values. The first term is \( k_1k_2k_3 \), the second term is \( k_2k_3k_4 \), and so on, until the last term \( k_{n-2}k_{n-1}k_n \).

### Step 2: Identifying overlapping terms
Notice that each term shares two of its variables with the next term. For example:
- The first term \( k_1k_2k_3 \) shares \( k_2 \) and \( k_3 \) with the second term \( k_2k_3k_4 \).
- Similarly, \( k_2k_3k_4 \) shares \( k_3 \) and \( k_4 \) with \( k_3k_4k_5 \), and so forth.

### Step 3: Setting up the equation
To satisfy the equation \( k_1k_2k_3 + k_2k_3k_4 + \ldots + k_{n-2}k_{n-1}k_n = 0 \), we can conclude that at least some of the \( k_i \) must be zero to ensure that the entire sum equals zero.

### Step 4: Finding the minimum number of zeros
To find the minimum number of zeros required, we can consider the impact of setting each \( k_i \) to zero:
- If we set \( k_3 = 0 \), then the terms \( k_1k_2k_3 \), \( k_2k_3k_4 \), and \( k_3k_4k_5 \) will all become zero. This means that three terms in the sum will be zero.
- Continuing this logic, if we set \( k_4 = 0 \), then \( k_2k_3k_4 \), \( k_3k_4k_5 \), and \( k_4k_5k_6 \) will also become zero, adding another three terms to the total.

### Step 5: Calculating the total number of terms
Since there are 12 terms in total, we can see that each zero can effectively nullify three terms. Therefore, to find the minimum number of zeros required, we can divide the total number of terms by the number of terms each zero can nullify:

\[ \text{Minimum number of zeros} = \frac{12}{3} = 4 \]

Thus, the minimum number of entities \( k_i \) that must be zero is **4**.

### Final Answer:
The minimum number of entities \( k_i \) that must be zero is **4**.