`""^(15)C_(0)*""^(5)C_(5)+""^(15)C_(1)*""^(5)C_(4)+""^(15)C_(2)*""^(5)C_(3)+""^(15)C_(3)*""^(5)C_(2)+""^(15)C_(4)*""^(5)C_(1)` is equal to :

Options

A. `2^(20)-2^(5)`
B. `(20!)/(5!15!)`
C. `(20!)/(5!15!)-1`
D. `(20!)/(5!15!)-(15!)/(5!10!)`

Answer

To solve the expression

\[ S = \sum_{k=0}^{4} \binom{15}{k} \binom{5}{5-k} \]

we can rewrite it as:

\[ S = \binom{15}{0} \binom{5}{5} + \binom{15}{1} \binom{5}{4} + \binom{15}{2} \binom{5}{3} + \binom{15}{3} \binom{5}{2} + \binom{15}{4} \binom{5}{1} \]

### Step 1: Recognize the Binomial Theorem
The expression can be interpreted using the binomial theorem. The binomial theorem states that:

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

### Step 2: Apply the Binomial Theorem
We can apply the binomial theorem to both \( (1+x)^{15} \) and \( (1+x)^{5} \):

\[ (1+x)^{15} = \sum_{k=0}^{15} \binom{15}{k} x^k \]
\[ (1+x)^{5} = \sum_{j=0}^{5} \binom{5}{j} x^j \]

### Step 3: Multiply the Two Expansions
Now, we multiply these two expansions:

\[ (1+x)^{15} \cdot (1+x)^{5} = (1+x)^{20} \]

### Step 4: Identify the Coefficient of \( x^5 \)
We need to find the coefficient of \( x^5 \) in the expansion of \( (1+x)^{20} \):

\[ (1+x)^{20} = \sum_{m=0}^{20} \binom{20}{m} x^m \]

The coefficient of \( x^5 \) is given by \( \binom{20}{5} \).

### Step 5: Include the Missing Term
However, in our original expression, we missed the term \( \binom{15}{5} \binom{5}{0} \). Therefore, we need to subtract this term from our result:

\[ S = \binom{20}{5} - \binom{15}{5} \]

### Step 6: Calculate the Values
Now we can calculate:

\[ \binom{20}{5} = \frac{20!}{5! \cdot 15!} \]
\[ \binom{15}{5} = \frac{15!}{5! \cdot 10!} \]

Thus,

\[ S = \frac{20!}{5! \cdot 15!} - \frac{15!}{5! \cdot 10!} \]

### Step 7: Simplify
Now we can simplify \( S \):

\[ S = \frac{20!}{5! \cdot 15!} - \frac{15!}{5! \cdot 10!} = \frac{20!}{5! \cdot 15!} - \frac{15! \cdot 15!}{5! \cdot 10! \cdot 15!} = \frac{20!}{5! \cdot 15!} - \frac{15!}{5! \cdot 10!} \]

### Final Result
This gives us the final answer for \( S \).