The coefficient of `x^(70)` in `x^2 (1 + x)^(98) + x^3 (1 + x)^(97) +x^4 (1 + x)^(96) +... + x^(54) (1 + x)^(46)` is `"^(99)C_p` – `"^(46)C_q`. Then a possible value of `p + q` is:

Options

A. 55
B. 61
C. 68
D. 83

Answer

To find the coefficient of \( x^{70} \) in the expression

\[ x^2 (1 + x)^{98} + x^3 (1 + x)^{97} + x^4 (1 + x)^{96} + \ldots + x^{54} (1 + x)^{46}, \]

we can rewrite this expression as a sum of terms. We can express it as follows:

\[ \sum_{k=2}^{54} x^k (1 + x)^{100 - k}. \]

### Step 1: Identify the general term

The general term in the sum can be written as:

\[ x^k (1 + x)^{100 - k}. \]

### Step 2: Find the coefficient of \( x^{70} \)

To find the coefficient of \( x^{70} \) in the entire expression, we need to consider each term \( x^k (1 + x)^{100 - k} \) and find the coefficient of \( x^{70 - k} \) in \( (1 + x)^{100 - k} \).

The coefficient of \( x^{70 - k} \) in \( (1 + x)^{100 - k} \) is given by the binomial coefficient:

\[ \binom{100 - k}{70 - k} = \binom{100 - k}{30}. \]

### Step 3: Set up the sum

Now, we can express the coefficient of \( x^{70} \) in the entire expression as:

\[ \sum_{k=2}^{54} \binom{100 - k}{30}. \]

### Step 4: Change the index of summation

To simplify the summation, we can change the index of summation by letting \( j = 100 - k \). When \( k = 2 \), \( j = 98 \) and when \( k = 54 \), \( j = 46 \). Thus, the sum becomes:

\[ \sum_{j=46}^{98} \binom{j}{30}. \]

### Step 5: Use the Hockey Stick Identity

Using the Hockey Stick Identity in combinatorics, we know that:

\[ \sum_{i=r}^{n} \binom{i}{r} = \binom{n+1}{r+1}. \]

Applying this identity, we have:

\[ \sum_{j=30}^{98} \binom{j}{30} = \binom{99}{31}. \]

### Step 6: Calculate the required sum

Now we need to subtract the terms from \( j = 30 \) to \( j = 45 \):

\[ \sum_{j=30}^{45} \binom{j}{30} = \binom{46}{31}. \]

Thus, the coefficient of \( x^{70} \) becomes:

\[ \binom{99}{31} - \binom{46}{31}. \]

### Step 7: Identify \( p \) and \( q \)

From the problem statement, we have:

\[ \binom{99}{p} - \binom{46}{q}. \]

By comparing, we can identify \( p = 31 \) and \( q = 31 \).

### Step 8: Find \( p + q \)

Finally, we calculate:

\[ p + q = 31 + 31 = 62. \]

Thus, the possible value of \( p + q \) is:

\[ \boxed{62}. \]