The number of rational terms in the binomial expansion of `(root(4)5+root(5)4)^(100)` is

Options

A. 50
B. 5
C. 6
D. 51

Answer

To find the number of rational terms in the binomial expansion of \((\sqrt{4}5 + \sqrt{5}4)^{100}\), we can follow these steps:

### Step 1: Identify the General Term
The general term \(T_r\) in the binomial expansion of \((a + b)^n\) is given by:
\[ T_r = \binom{n}{r} a^{n-r} b^r \]
In our case, \(a = \sqrt{4}5\) and \(b = \sqrt{5}4\), and \(n = 100\). Therefore, the general term can be expressed as:
\[ T_r = \binom{100}{r} (\sqrt{4}5)^{100-r} (\sqrt{5}4)^r \]

### Step 2: Simplify the General Term
We can simplify the general term:
\[ T_r = \binom{100}{r} (2 \cdot 5)^{100-r} (2 \cdot \sqrt{5})^r \]
This simplifies to:
\[ T_r = \binom{100}{r} 2^{100} 5^{100-r} 2^r (\sqrt{5})^r \]
Thus, we have:
\[ T_r = \binom{100}{r} 2^{100+r} 5^{(100-r)/2} 5^{r/2} \]
Combining the powers of \(5\):
\[ T_r = \binom{100}{r} 2^{100+r} 5^{(100+r)/2} \]

### Step 3: Determine Conditions for Rational Terms
For \(T_r\) to be a rational term, both \(100 - r\) must be even (to ensure \((100 - r)/2\) is an integer) and \(r\) must also be even (to ensure \(r/2\) is an integer).

### Step 4: Set Up the Conditions
1. \(100 - r\) is even: This implies \(r\) must be even.
2. \(r\) is even: This means \(r\) can take values from \(0\) to \(100\) in steps of \(2\).

### Step 5: Find Possible Values for \(r\)
Since \(r\) must be a multiple of both \(4\) and \(5\), the least common multiple (LCM) of \(4\) and \(5\) is \(20\). Therefore, \(r\) can take values:
\[ r = 0, 20, 40, 60, 80, 100 \]
This gives us the possible values of \(r\) as \(0, 20, 40, 60, 80, 100\).

### Step 6: Count the Rational Terms
The values of \(r\) that satisfy the conditions are \(0, 20, 40, 60, 80, 100\). This gives us a total of \(6\) rational terms.

### Final Answer
Thus, the number of rational terms in the binomial expansion of \((\sqrt{4}5 + \sqrt{5}4)^{100}\) is \(6\).