In the expansion of `(1+x)^(2)(1+y)^(3)(1+z)^(4)(1+w)^(5)`, the sum of the coefficient of the terms of degree 12 is :

Options

A. 61
B. 71
C. 81
D. 91

Answer

To find the sum of the coefficients of the terms of degree 12 in the expansion of \((1+x)^{2}(1+y)^{3}(1+z)^{4}(1+w)^{5}\), we can follow these steps:

### Step 1: Understand the Expansion
The expression can be rewritten as:
\[ (1+x)^{2} \cdot (1+y)^{3} \cdot (1+z)^{4} \cdot (1+w)^{5} \]
We need to find the coefficient of \(x^{12}\) in this expansion.

### Step 2: Combine the Exponents
The total degree of the terms in the expansion is given by the sum of the exponents from each binomial expansion. The maximum degree we can achieve from each binomial is:
- From \((1+x)^{2}\): maximum degree is 2
- From \((1+y)^{3}\): maximum degree is 3
- From \((1+z)^{4}\): maximum degree is 4
- From \((1+w)^{5}\): maximum degree is 5

The total maximum degree is:
\[ 2 + 3 + 4 + 5 = 14 \]

### Step 3: Find the Coefficient of \(x^{12}\)
We need to find the coefficient of \(x^{12}\) in the expansion. This means we want to find the combinations of terms from each binomial that sum to 12.

To find the coefficient of \(x^{12}\), we can consider the overall expansion as:
\[ (1+x+y+z+w)^{14} \]
We need to find the coefficient of \(x^{12}\) in this expansion.

### Step 4: Use the Binomial Theorem
Using the multinomial expansion, we can express the coefficient of \(x^{12}\) as:
\[ \text{Coefficient of } x^{12} = \text{Coefficient of } x^{12} \text{ in } (1+x+y+z+w)^{14} \]
This can be calculated using the multinomial coefficient:
\[ \frac{14!}{12! \cdot 0! \cdot 0! \cdot 0! \cdot 2!} \]
where \(12\) is the power of \(x\) and \(2\) is the remaining degree which can be distributed among \(y\), \(z\), and \(w\).

### Step 5: Calculate the Coefficient
Calculating the multinomial coefficient:
\[ \frac{14!}{12! \cdot 2!} = \frac{14 \times 13}{2 \times 1} = 91 \]

### Conclusion
The sum of the coefficients of the terms of degree 12 in the expansion of \((1+x)^{2}(1+y)^{3}(1+z)^{4}(1+w)^{5}\) is **91**.