Using binomial theorem, expand `[ ( x+y)^(5) + (x-y)^(5) ]` and hence find the value of `[ ( sqrt(3) + 1)^(5) - ( sqrt(3) - 1)^(5) ]`.

Answer

To solve the problem, we will first expand \((x+y)^5 + (x-y)^5\) using the Binomial Theorem and then substitute specific values to find the required expression.

### Step 1: Expand \((x+y)^5\) using the Binomial Theorem

According to the Binomial Theorem, we have:
\[ (x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \]
For \(n=5\), the expansion of \((x+y)^5\) is:
\[ (x+y)^5 = \binom{5}{0} x^5 y^0 + \binom{5}{1} x^4 y^1 + \binom{5}{2} x^3 y^2 + \binom{5}{3} x^2 y^3 + \binom{5}{4} x^1 y^4 + \binom{5}{5} x^0 y^5 \]
Calculating the binomial coefficients:
\[ = 1 \cdot x^5 + 5 \cdot x^4 y + 10 \cdot x^3 y^2 + 10 \cdot x^2 y^3 + 5 \cdot x y^4 + 1 \cdot y^5 \]
Thus,
\[ (x+y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5 \]

### Step 2: Expand \((x-y)^5\) using the Binomial Theorem

Now, we expand \((x-y)^5\):
\[ (x-y)^5 = \sum_{k=0}^{5} \binom{5}{k} x^{5-k} (-y)^k \]
This gives:
\[ = \binom{5}{0} x^5 (-y)^0 + \binom{5}{1} x^4 (-y)^1 + \binom{5}{2} x^3 (-y)^2 + \binom{5}{3} x^2 (-y)^3 + \binom{5}{4} x^1 (-y)^4 + \binom{5}{5} x^0 (-y)^5 \]
Calculating the terms:
\[ = x^5 - 5x^4y + 10x^3y^2 - 10x^2y^3 + 5xy^4 - y^5 \]

### Step 3: Add the two expansions

Now we add \((x+y)^5\) and \((x-y)^5\):
\[ (x+y)^5 + (x-y)^5 = (x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5) + (x^5 - 5x^4y + 10x^3y^2 - 10x^2y^3 + 5xy^4 - y^5) \]
Combining like terms:
\[ = 2x^5 + 0 + 20x^3y^2 + 0 + 10xy^4 + 0 \]
Thus, we have:
\[ (x+y)^5 + (x-y)^5 = 2x^5 + 20x^3y^2 + 10xy^4 \]

### Step 4: Substitute \(x = \sqrt{3}\) and \(y = 1\)

Now we substitute \(x = \sqrt{3}\) and \(y = 1\) into the expression:
\[ 2(\sqrt{3})^5 + 20(\sqrt{3})^3(1)^2 + 10(\sqrt{3})(1)^4 \]
Calculating each term:
1. \((\sqrt{3})^5 = 3^{5/2} = 3^2 \cdot \sqrt{3} = 9\sqrt{3}\)
2. \((\sqrt{3})^3 = 3^{3/2} = 3\sqrt{3}\)

Substituting these values:
\[ = 2(9\sqrt{3}) + 20(3\sqrt{3}) + 10(\sqrt{3}) \]
\[ = 18\sqrt{3} + 60\sqrt{3} + 10\sqrt{3} \]
Combining these:
\[ = (18 + 60 + 10)\sqrt{3} = 88\sqrt{3} \]

### Final Answer
Thus, the value of \((\sqrt{3}+1)^5 - (\sqrt{3}-1)^5\) is:
\[ \boxed{88\sqrt{3}} \]