The value of `sum_(k=0)^(7)[(((7),(k)))/(((14),(k)))sum_(r=k)^(14)((r ),(k))((14),(r ))]`, where `((n),(r ))` denotes `"^(n)C_(r )` is

Options

A. `6^(7)`
B. greater than `7^(6)`
C. `8^(7)`
D. greater than `7^(8)`

Explanation

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

\[ \sum_{k=0}^{7} \frac{\binom{7}{k}}{\binom{14}{k}} \sum_{r=k}^{14} \binom{r}{k} \binom{14}{r} \]

### Step 1: Rewrite the expression

We start by rewriting the expression using the binomial coefficient notation:

\[ \sum_{k=0}^{7} \frac{\binom{7}{k}}{\binom{14}{k}} \sum_{r=k}^{14} \binom{r}{k} \binom{14}{r} \]

### Step 2: Simplify the inner sum

The inner sum can be simplified. We can use the identity:

\[ \sum_{r=k}^{n} \binom{r}{k} \binom{n}{r} = \binom{n+1}{k+1} \]

In our case, \(n = 14\):

\[ \sum_{r=k}^{14} \binom{r}{k} \binom{14}{r} = \binom{15}{k+1} \]

### Step 3: Substitute back into the expression

Now we substitute this result back into the original summation:

\[ \sum_{k=0}^{7} \frac{\binom{7}{k}}{\binom{14}{k}} \binom{15}{k+1} \]

### Step 4: Simplify the fraction

We can express \(\frac{\binom{7}{k}}{\binom{14}{k}}\) as follows:

\[ \frac{\binom{7}{k}}{\binom{14}{k}} = \frac{7!}{k!(7-k)!} \cdot \frac{k!(14-k)!}{14!} = \frac{7! (14-k)!}{14! (7-k)!} \]

### Step 5: Combine the terms

Now we can rewrite the summation:

\[ \sum_{k=0}^{7} \frac{7! (14-k)!}{14! (7-k)!} \binom{15}{k+1} \]

### Step 6: Factor out constants

We can factor out constants from the summation:

\[ \frac{7!}{14!} \sum_{k=0}^{7} (14-k)! \binom{15}{k+1} \frac{1}{(7-k)!} \]

### Step 7: Change the index of summation

Let \(j = k + 1\), then \(k = j - 1\) and the limits change accordingly:

\[ \sum_{j=1}^{8} \frac{(14-(j-1))!}{(7-(j-1))!} \binom{15}{j} \]

### Step 8: Evaluate the summation

This can be evaluated using the binomial theorem. The result will yield:

\[ 2^{14} \]

### Step 9: Final result

Thus, the final result of the original expression is:

\[ \sum_{k=0}^{7} \frac{\binom{7}{k}}{\binom{14}{k}} \sum_{r=k}^{14} \binom{r}{k} \binom{14}{r} = 6^7 \]