Number of non-empty subsets `{1,2,3,4,5,6,7,8}` having exactly k elements and do not contain the element k for some `k = 1,2"….."8` is

Options

A. 63
B. 255
C. 127
D. 31

Answer

To solve the problem of finding the number of non-empty subsets of the set `{1, 2, 3, 4, 5, 6, 7, 8}` that have exactly `k` elements and do not contain the element `k`, we can follow these steps:

### Step 1: Understand the Problem
We need to find subsets of size `k` from the set `{1, 2, 3, 4, 5, 6, 7, 8}` such that the subset does not include the element `k`.

### Step 2: Determine the Available Elements
If we are not including the element `k`, then we are left with the elements `{1, 2, 3, 4, 5, 6, 7, 8} \ {k}`. This leaves us with 7 elements.

### Step 3: Calculate the Number of Subsets for Each `k`
For each `k` from 1 to 7, the number of ways to choose `k` elements from the remaining 7 elements can be calculated using the binomial coefficient `C(7, k)`.

- For `k = 1`: We choose 1 element from 7, which is `C(7, 1)`.
- For `k = 2`: We choose 2 elements from 7, which is `C(7, 2)`.
- For `k = 3`: We choose 3 elements from 7, which is `C(7, 3)`.
- For `k = 4`: We choose 4 elements from 7, which is `C(7, 4)`.
- For `k = 5`: We choose 5 elements from 7, which is `C(7, 5)`.
- For `k = 6`: We choose 6 elements from 7, which is `C(7, 6)`.
- For `k = 7`: We choose 7 elements from 7, which is `C(7, 7)`.

### Step 4: Sum the Binomial Coefficients
Now we need to sum these binomial coefficients:
\[ \text{Total} = C(7, 1) + C(7, 2) + C(7, 3) + C(7, 4) + C(7, 5) + C(7, 6) + C(7, 7) \]

### Step 5: Use the Binomial Theorem
According to the binomial theorem, the sum of the binomial coefficients for a given `n` is:
\[ \sum_{i=0}^{n} C(n, i) = 2^n \]
Thus, for `n = 7`:
\[ \sum_{i=0}^{7} C(7, i) = 2^7 = 128 \]

### Step 6: Exclude the Empty Subset
Since we are interested in non-empty subsets, we need to subtract the empty subset `C(7, 0) = 1` from our total:
\[ \text{Total non-empty subsets} = 128 - 1 = 127 \]

### Conclusion
The total number of non-empty subsets of the set `{1, 2, 3, 4, 5, 6, 7, 8}` having exactly `k` elements and not containing the element `k` is **127**.