Let `(1+x)^(10) = sum_(r=0)^(10) c_(r) x^ (r ) and (1+x)^(7) = sum_(r=0)^(7) d_(r) x^(r )`.
If `P= sum_(r=0)^(5) c_(2r) and Q= sum_(r=0)^(3) d_(2r+1),` then `(P)/(Q)` is equal to

Options

A. 4
B. 8
C. 16
D. 32

Answer

To solve the problem, we need to find the values of \( P \) and \( Q \) based on the given binomial expansions, and then compute \( \frac{P}{Q} \).

### Step 1: Calculate \( P \)

We start with the expression for \( P \):
\[ P = \sum_{r=0}^{5} c_{2r} \]
Here, \( c_r \) are the coefficients from the expansion of \( (1+x)^{10} \).

Using the binomial theorem:
\[ (1+x)^{10} = \sum_{r=0}^{10} c_r x^r \]
To find \( P \), we can evaluate \( (1+x)^{10} \) at \( x=1 \) and \( x=-1 \):
1. At \( x = 1 \):
\[ (1+1)^{10} = 2^{10} = 1024 \]
This gives us the sum of all coefficients:
\[ c_0 + c_1 + c_2 + \ldots + c_{10} = 1024 \]

2. At \( x = -1 \):
\[ (1-1)^{10} = 0 \]
This gives us:
\[ c_0 - c_1 + c_2 - c_3 + c_4 - c_5 + c_6 - c_7 + c_8 - c_9 + c_{10} = 0 \]

From these two equations, we can derive:
\[ c_0 + c_2 + c_4 + c_6 + c_8 + c_{10} = \frac{1024}{2} = 512 \]
This is the sum of the coefficients of even powers, which is equal to \( P \):
\[ P = 2^{9} = 512 \]

### Step 2: Calculate \( Q \)

Next, we compute \( Q \):
\[ Q = \sum_{r=0}^{3} d_{2r+1} \]
Here, \( d_r \) are the coefficients from the expansion of \( (1+x)^{7} \).

Using the binomial theorem:
\[ (1+x)^{7} = \sum_{r=0}^{7} d_r x^r \]
We evaluate \( (1+x)^{7} \) at \( x=1 \) and \( x=-1 \):
1. At \( x = 1 \):
\[ (1+1)^{7} = 2^{7} = 128 \]
This gives us:
\[ d_0 + d_1 + d_2 + \ldots + d_{7} = 128 \]

2. At \( x = -1 \):
\[ (1-1)^{7} = 0 \]
This gives us:
\[ d_0 - d_1 + d_2 - d_3 + d_4 - d_5 + d_6 - d_7 = 0 \]

From these two equations, we can derive:
\[ d_0 + d_2 + d_4 + d_6 = \frac{128}{2} = 64 \]
And,
\[ d_1 + d_3 + d_5 + d_7 = \frac{128}{2} = 64 \]
Thus, we find:
\[ Q = d_1 + d_3 + d_5 + d_7 = 64 \]

### Step 3: Calculate \( \frac{P}{Q} \)

Now we can compute \( \frac{P}{Q} \):
\[ \frac{P}{Q} = \frac{512}{64} = 8 \]

### Final Result

Thus, the final answer is:
\[ \frac{P}{Q} = 8 \]