The number `N=""^(20)C_(7)-""^(20)C_(8)+""^(20)C_(9)-""^(20)C_(10)+….. -""^(20)C_(20)` is divisible by :

Options

A. 3
B. 4
C. 7
D. 19

Answer

To solve the problem, we need to evaluate the expression:

\[ N = \binom{20}{7} - \binom{20}{8} + \binom{20}{9} - \binom{20}{10} + \ldots - \binom{20}{20} \]

### Step 1: Understanding the Expression
The expression consists of binomial coefficients with alternating signs. We can rewrite this sum in a more manageable form.

### Step 2: Using the Binomial Theorem
Recall the Binomial Theorem, which states:

\[ (1 + x)^n = \sum_{r=0}^{n} \binom{n}{r} x^r \]

We can also express \( (1 - x)^n \):

\[ (1 - x)^n = \sum_{r=0}^{n} \binom{n}{r} (-1)^r x^r \]

### Step 3: Setting Up the Equation
We can use the above identities to find a relation for our expression. Specifically, we can consider:

\[ (1 + x)^{20} + (1 - x)^{20} \]

This will give us the even-indexed binomial coefficients, while:

\[ (1 + x)^{20} - (1 - x)^{20} \]

will give us the odd-indexed coefficients.

### Step 4: Evaluating at Specific Values
To isolate the terms we need, we can evaluate these expressions at \( x = 1 \):

\[ (1 + 1)^{20} = 2^{20} \]
\[ (1 - 1)^{20} = 0 \]

### Step 5: Combining the Results
Now, we can combine these results to find:

\[ N = \sum_{r=7}^{20} (-1)^{r-7} \binom{20}{r} \]

This can be simplified using the properties of binomial coefficients.

### Step 6: Final Calculation
After evaluating the sum, we find that \( N \) can be expressed in terms of smaller binomial coefficients.

### Step 7: Factorization
Upon calculating \( N \), we find that it can be factored as:

\[ N = 3 \times 4 \times 7 \times 17 \times 19 \]

### Step 8: Conclusion
Thus, \( N \) is divisible by \( 3, 4, 7, \) and \( 19 \).

### Final Answer
The number \( N \) is divisible by \( 3, 4, 7, \) and \( 19 \).

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