Evaluate`sum_(i=0)^(n)sum_(j=0)^(n)sum_(k=0)^(n)((n),(i))((n),(j))((n),(k)), where((n),(r))=""^(n)C_(r)`: (a) is less than 500 if n = 3 (b) is greater than 600 if n = 3 (c) is less than 5000 if n = 4 (d) is greater than 4000 if n = 4

Options

A. is less than 500 if n = 3
B. is greater than 600 if n = 3
C. is less than 5000 if n = 4
D. is greater than 4000 if n = 4

Answer

To evaluate the expression

\[ \sum_{i=0}^{n} \sum_{j=0}^{n} \sum_{k=0}^{n} \binom{n}{i} \binom{n}{j} \binom{n}{k}, \]

we can follow these steps:

### Step 1: Rewrite the Expression
We start by recognizing that the summation over \(k\) can be factored out since \(\binom{n}{i}\) and \(\binom{n}{j}\) are constants with respect to \(k\):

\[ \sum_{i=0}^{n} \sum_{j=0}^{n} \binom{n}{i} \binom{n}{j} \sum_{k=0}^{n} \binom{n}{k}. \]

### Step 2: Evaluate the Innermost Sum
Next, we evaluate the innermost sum:

\[ \sum_{k=0}^{n} \binom{n}{k} = 2^n. \]

This follows from the binomial theorem, which states that the sum of the binomial coefficients for a given \(n\) equals \(2^n\).

### Step 3: Substitute Back into the Expression
Substituting back, we have:

\[ \sum_{i=0}^{n} \sum_{j=0}^{n} \binom{n}{i} \binom{n}{j} \cdot 2^n. \]

### Step 4: Factor Out the Constant
Now we can factor out \(2^n\):

\[ 2^n \sum_{i=0}^{n} \sum_{j=0}^{n} \binom{n}{i} \binom{n}{j}. \]

### Step 5: Evaluate the Remaining Double Sum
The remaining double sum can be simplified as follows:

\[ \sum_{i=0}^{n} \binom{n}{i} \sum_{j=0}^{n} \binom{n}{j} = \left(\sum_{i=0}^{n} \binom{n}{i}\right) \left(\sum_{j=0}^{n} \binom{n}{j}\right) = 2^n \cdot 2^n = 2^{2n}. \]

### Step 6: Combine Everything
Now we combine everything:

\[ 2^n \cdot 2^{2n} = 2^{3n}. \]

### Step 7: Evaluate for Specific Values of \(n\)
Now we can evaluate this for \(n = 3\) and \(n = 4\):

1. For \(n = 3\):
\[ 2^{3 \cdot 3} = 2^9 = 512. \]

2. For \(n = 4\):
\[ 2^{3 \cdot 4} = 2^{12} = 4096. \]

### Step 8: Check the Options
Now we can check the options given in the question:

- For \(n = 3\), \(512\) is:
- (a) less than 500: **False**
- (b) greater than 600: **False**

- For \(n = 4\), \(4096\) is:
- (c) less than 5000: **True**
- (d) greater than 4000: **True**

### Conclusion
Thus, the correct options are (c) and (d).