The number `N=""^(20)C_(7)-""^(20)C_(8)+""^(20)C_(9)-""^(20)C_(10)+….. -""^(20)C_(20)` is not divisible by :

Options

A. 3
B. 7
C. 11
D. 19

Answer

To solve the problem, we need to evaluate the expression

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

and determine which number it is not divisible by.

### Step 1: Understanding the Expression

The expression can be rewritten using the binomial theorem. We know that:

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

For our case, we can use \( x = -1 \) to find the alternating sum:

\[ (1 - 1)^{20} = \sum_{r=0}^{20} \binom{20}{r} (-1)^r = 0 \]

This means that the sum of all the binomial coefficients from \( \binom{20}{0} \) to \( \binom{20}{20} \) is zero.

### Step 2: Splitting the Expression

We can split the expression into even and odd indexed terms:

\[ N = \left( \binom{20}{7} + \binom{20}{9} + \binom{20}{11} + \ldots + \binom{20}{19} \right) - \left( \binom{20}{8} + \binom{20}{10} + \ldots + \binom{20}{20} \right) \]

### Step 3: Using Binomial Coefficients Properties

We can also use the identity:

\[ \binom{n}{r} = \binom{n}{n-r} \]

This allows us to rewrite the terms. For instance:

\[ \binom{20}{8} = \binom{20}{12}, \quad \binom{20}{9} = \binom{20}{11}, \quad \text{and so on.} \]

### Step 4: Evaluating the Expression

Using the property of binomial coefficients, we can express \( N \) as follows:

\[ N = \sum_{k=0}^{6} \left( \binom{20}{7+k} - \binom{20}{8+k} \right) \]

### Step 5: Calculate \( N \)

We can compute \( N \) directly:

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

Calculating these values, we find:

\[ \begin{align*} \binom{20}{7} & = 77520 \\ \binom{20}{8} & = 125970 \\ \binom{20}{9} & = 167960 \\ \binom{20}{10} & = 184756 \\ \ldots & \\ \binom{20}{20} & = 1 \\ \end{align*} \]

After performing the calculations, we find:

\[ N = 27132 \]

### Step 6: Check Divisibility

Now we need to check the divisibility of \( 27132 \) by the given options: \( 3, 7, 11, 19 \).

- **Divisibility by 3**: \( 27132 \div 3 = 9044 \) (divisible)
- **Divisibility by 7**: \( 27132 \div 7 = 3876 \) (divisible)
- **Divisibility by 19**: \( 27132 \div 19 = 1422 \) (divisible)
- **Divisibility by 11**: \( 27132 \div 11 = 2466.545 \) (not divisible)

### Conclusion

The number \( N = 27132 \) is **not divisible by 11**.