Expand (x-1/2y)^5 by using combination method
To expand (x - 1/2y)^5 using the combination method, we can use the binomial theorem. The binomial theorem states that (a + b)^n = C(n, 0)a^n*b^0 + C(n, 1)a^(n-1)*b^1 + C(n, 2)a^(n-2)*b^2 + ... + C(n, n-1)a^1*b^(n-1) + C(n, n)a^0*b^n, where C(n, k) is the combination formula which calculates the number of ways to choose k items from a set of n items.
In this case, a = x and b = -1/2y. So, the expansion of (x - 1/2y)^5 will be:
C(5, 0)x^5(-1/2y)^0 + C(5, 1)x^4(-1/2y)^1 + C(5, 2)x^3(-1/2y)^2 + C(5, 3)x^2(-1/2y)^3 + C(5, 4)x^1(-1/2y)^4 + C(5, 5)x^0(-1/2y)^5
Now, let's calculate each term:
C(5, 0)x^5(-1/2y)^0 = 1*x^5*(-1/2y)^0 = x^5
C(5, 1)x^4(-1/2y)^1 = 5*x^4*(-1/2y)^1 = -5/2*x^4y
C(5, 2)x^3(-1/2y)^2 = 10*x^3*(-1/2y)^2 = 10/4*x^3y^2 = 5/2*x^3y^2
C(5, 3)x^2(-1/2y)^3 = 10*x^2*(-1/2y)^3 = -10/8*x^2y^3 = -5/4*x^2y^3
C(5, 4)x^1(-1/2y)^4 = 5*x^1*(-1/2y)^4 = 5/16*x^1y^4
C(5, 5)x^0(-1/2y)^5 = 1*x^0*(-1/2y)^5 = -1/32*y^5
Putting it all together, the expansion of (x - 1/2y)^5 is:
x^5 - 5/2*x^4y + 5/2*x^3y^2 - 5/4*x^2y^3 + 5/16*x^1y^4 - 1/32*y^5
To expand (x - (1/2)y)^5 using the combination method, we need to utilize the binomial theorem.
The binomial theorem states that for any binomial expression (a + b)^n, where n is a positive integer, the expansion is given by:
(a + b)^n = C(n, 0) * a^(n-0) * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n-1) * a^(1) * b^(n-1) + C(n, n) * a^(0) * b^n
where C(n, k) denotes the binomial coefficient, which is defined as:
C(n, k) = n! / (k! * (n - k)!)
Now let's expand (x - (1/2)y)^5:
Using the binomial theorem, we can write the expansion as:
C(5, 0) * x^5 * (-1/2y)^0 + C(5, 1) * x^4 * (-1/2y)^1 + C(5, 2) * x^3 * (-1/2y)^2 + C(5, 3) * x^2 * (-1/2y)^3 + C(5, 4) * x^1 * (-1/2y)^4 + C(5, 5) * x^0 * (-1/2y)^5
Simplifying each term and expanding the binomial coefficients, we get:
1 * x^5 * 1 + 5 * x^4 * (-1/2y) + 10 * x^3 * (-1/2y)^2 + 10 * x^2 * (-1/2y)^3 + 5 * x^1 * (-1/2y)^4 + 1 * (-1/2y)^5
Simplifying further, we have:
x^5 - 5/2 * x^4 * y + 5/4 * x^3 * y^2 - 5/8 * x^2 * y^3 + 5/16 * x * y^4 - 1/32 * y^5
Therefore, the expanded form of (x - (1/2)y)^5 using the combination method is:
x^5 - 5/2 * x^4 * y + 5/4 * x^3 * y^2 - 5/8 * x^2 * y^3 + 5/16 * x * y^4 - 1/32 * y^5