The expression
`""^(n)C_(r)+4.""^(n)C_(r-1)+6.""^(n)C_(r-2)+4.""^(n)C_(r-3)+""^(n)C_(r-4)`

Options

A. `((n+4),(r+4))`
B. `((n+4),(r))`
C. `((n+3),(r-1))`
D. `((n+4)/(r+3))`

Answer

To solve the expression

\[ \binom{n}{r} + 4\binom{n}{r-1} + 6\binom{n}{r-2} + 4\binom{n}{r-3} + \binom{n}{r-4} \]

we can use the properties of binomial coefficients.

### Step 1: Group the terms

We can group the terms in pairs to utilize the identity:

\[ \binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k} \]

### Step 2: Rewrite the expression

We can rewrite the expression as follows:

\[ \binom{n}{r} + \binom{n}{r-4} + 4\left(\binom{n}{r-1} + \binom{n}{r-3}\right) + 2\binom{n}{r-2} \]

### Step 3: Apply the identity

Now, we apply the identity to the pairs:

1. For \(\binom{n}{r} + \binom{n}{r-4}\):
\[ \binom{n}{r} + \binom{n}{r-4} = \binom{n+1}{r-1} + \binom{n+1}{r-3} \]

2. For \(4\left(\binom{n}{r-1} + \binom{n}{r-3}\right)\):
\[ 4\left(\binom{n}{r-1} + \binom{n}{r-3}\right) = 4\binom{n+1}{r-2} \]

3. For \(2\binom{n}{r-2}\):
\[ 2\binom{n}{r-2} \]

### Step 4: Combine the results

Now we combine the results:

\[ \binom{n+1}{r-1} + 4\binom{n+1}{r-2} + \binom{n+1}{r-3} \]

### Step 5: Apply the identity again

We can apply the identity again to combine these terms:

\[ \binom{n+1}{r-1} + \binom{n+1}{r-3} = \binom{n+2}{r-2} \]

### Step 6: Final expression

Thus, we can express the entire original expression as:

\[ \binom{n+4}{r} \]

### Conclusion

The final result is:

\[ \binom{n+4}{r} \]