Question

The total number of terms in the expansion of $ (x + y)^{60} + (x - y)^{60} $ is

• A. 60
• B. 61
• C. 30
• D. 31

Hint:

When adding expansions of binomial expressions, terms with the same powers of \(x\) and \(y\) (even powers in this case) will remain, while terms with odd powers of \(y\) will cancel out.

Answer 1

The Correct Option is D

To determine the total number of unique terms in the expansion of $ (x + y)^{60} + (x - y)^{60} $, we can approach the problem as follows:

The binomial expansion of $(x+y)^{60}$ and $(x-y)^{60}$ are:

• $(x+y)^{60}$: $ \sum_{k=0}^{60} \binom{60}{k} x^{60-k} y^{k} $
• $(x-y)^{60}$: $ \sum_{k=0}^{60} \binom{60}{k} x^{60-k} (-y)^{k}$, which is $ \sum_{k=0}^{60} \binom{60}{k} x^{60-k} (-1)^{k} y^{k} $

The terms in these expansions depend on the parity of $k$ (even or odd):

• For $k$ even: Both $(x+y)^{60}$ and $(x-y)^{60}$ contribute the terms $2 \binom{60}{k} x^{60-k} y^{k}$.
• For $k$ odd: The terms cancel each other out because of the opposite signs from $(x+y)^{60}$ and $(x-y)^{60}$.

This results in non-zero terms only when $k$ is even. Since $k$ can go from 0 to 60, the sequence of even numbers between 0 and 60 is $0, 2, 4, ..., 60$.

The number of even numbers in this sequence can be found by computing the number of terms in an arithmetic sequence:

The sequence has a common difference of 2. The formula for the number of terms $n$ in an arithmetic sequence is calculated as:

• $n = \frac{\text{Last term} - \text{First term}}{\text{Common difference}} + 1$

Substituting the relevant values, we find:

• $n = \frac{60 - 0}{2} + 1 = 31$

Thus, the total number of terms in the expansion of $ (x + y)^{60} + (x - y)^{60}$ is 31.

Answer 2

The number of terms in the expansion of \( (x + y)^{60} \) is given by the binomial expansion formula: \[ \text{Number of terms in } (x + y)^{60} = 60 + 1 = 61 \] 
Similarly, the number of terms in the expansion of \( (x - y)^{60} \) is also: \[ \text{Number of terms in } (x - y)^{60} = 60 + 1 = 61 \] 
Now, we need to consider the combined expansion of both \( (x + y)^{60} \) and \( (x - y)^{60} \). 
In both expansions, terms with odd powers of \(y\) will cancel out because of the signs in \( (x - y)^{60} \), and terms with even powers of \(y\) will remain. 
Therefore, only the even-powered terms from both expansions will contribute, which will give: \[ \text{Total number of terms} = \frac{60}{2} + 1 = 31 \] 
Thus, the total number of terms is \(31\).