N.O.T in the expansion (x+a)^n -(x-a)^n=

Number of Terms in the Expansion of (x^4)^31 - (x-4)^31

To find the number of terms in the expansion of the given expression, we need to first expand each term separately and then subtract them.

Expanding (x^4)^31:
When raising a power to another power, we multiply the exponents. Therefore, expanding (x^4)^31 results in x^(4*31) = x^124.

Expanding (x-4)^31:
To expand (x-4)^31, we can use the binomial expansion formula, which is:

(x - y)^n = C(n, 0)x^n * y^0 + C(n, 1)x^(n-1) * y^1 + C(n, 2)x^(n-2) * y^2 + ... + C(n, n-1)x * y^(n-1) + C(n, n)y^n

In this case, x is the variable x and y is -4. The term C(n, k) represents the binomial coefficient, which is calculated using the formula:

C(n, k) = n! / (k! * (n-k)!)

Using the binomial expansion formula, we can expand (x-4)^31 and determine the number of terms.

Calculating the Number of Terms:
Since (x^4)^31 is a monomial, it consists of only one term: x^124.

Now, let's determine the number of terms in the expansion of (x-4)^31. We will substitute the values into the binomial expansion formula and calculate each term individually.

The binomial coefficient C(n, k) can be calculated as follows:

C(31, k) = 31! / (k! * (31-k)!)

We need to calculate the number of terms for k ranging from 0 to 31 and sum them up.

The number of terms in the expansion of (x-4)^31 can be calculated using the formula:

Number of Terms = ∑ C(31, k) for k = 0 to 31

Substituting the values, we can calculate the number of terms.

Finally, subtract the number of terms in (x^4)^31 from the number of terms in (x-4)^31 to find the number of terms in the expression (x^4)^31 - (x-4)^31.

Example:
Let's consider the expansion of (x-4)^3 as an example to illustrate the process.

Using the binomial expansion formula, we have:

(x-4)^3 = C(3, 0)x^3 * (-4)^0 + C(3, 1)x^2 * (-4)^1 + C(3, 2)x^1 * (-4)^2 + C(3, 3)x^0 * (-4)^3

Expanding and simplifying each term, we get:

(x-4)^3 = x^3 - 12x^2 + 48x - 64

In this case, there are four terms in the expansion of (x-4)^3.

By following the same process for the given expression ((x^4)^31 - (x-4)^31), we can find the