Find `14C_1 +14 C_2 + 14 C_3 +….. + 14 C_14=` ?

Answer

To solve the problem of finding the sum \( 14C_1 + 14C_2 + 14C_3 + \ldots + 14C_{14} \), we can use a property of binomial coefficients. Here’s a step-by-step solution:

### Step 1: Understand the Binomial Coefficient Sum
The sum of the binomial coefficients from \( nC_0 \) to \( nC_n \) is given by the formula:
\[ \sum_{k=0}^{n} nC_k = 2^n \]
This means that the total sum of all binomial coefficients for a given \( n \) equals \( 2^n \).

### Step 2: Apply the Formula
In our case, we want to find the sum from \( 14C_1 \) to \( 14C_{14} \). Notice that:
\[ \sum_{k=0}^{14} 14C_k = 2^{14} \]
This includes \( 14C_0 \) as well, which is equal to 1. Therefore, we can express our desired sum as:
\[ 14C_1 + 14C_2 + 14C_3 + \ldots + 14C_{14} = \sum_{k=0}^{14} 14C_k - 14C_0 \]

### Step 3: Calculate \( 2^{14} \)
Now, we calculate \( 2^{14} \):
\[ 2^{14} = 16384 \]

### Step 4: Subtract \( 14C_0 \)
Since \( 14C_0 = 1 \), we subtract this from the total:
\[ 14C_1 + 14C_2 + 14C_3 + \ldots + 14C_{14} = 16384 - 1 = 16383 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{16383} \]