Find the seventh term of the expansion ( x + y)10
210x^4y^6
Sure, let me calculate that for you. Give me a moment...
*thinking*
Hmm, you got a multiplication going on there. Okay, let's clown around with it.
It seems like you're expanding the expression (x + y) raised to the power of 10. Now, let's skip the first six terms because we're looking for the seventh term.
To calculate the seventh term, we have to look at the coefficients of the terms and reminisce about Pascal's Triangle. In this case, starting with the term (x^10) and (y^0), the coefficients follow this pattern: 1, 10, 45, 120, 210, 252.
So, the seventh term would be 252 * (x^4) * (y^6). Voila! That's your answer. Hope that brings a smile to your face!
To find the seventh term of the expansion (x + y)^10, we can use the binomial theorem.
The general formula for finding the kth term of the binomial expansion (x + y)^n is:
C(n, k) * x^(n-k) * y^k
where C(n, k) represents the binomial coefficient.
In our case, n = 10 and we want to find the seventh term, so k = 7.
Plugging these values into the formula, we get:
C(10, 7) * x^(10-7) * y^7
To calculate C(10, 7), we use the combination formula: C(n, k) = n! / (k! * (n-k)!)
C(10, 7) = 10! / (7! * (10-7)!)
Simplifying this expression:
C(10, 7) = 10! / (7! * 3!) = (10 * 9 * 8) / (3 * 2 * 1) = 120
Now we can substitute this value back into the formula:
120 * x^(10-7) * y^7
Simplifying further:
120 * x^3 * y^7
Therefore, the seventh term of the expansion (x + y)^10 is 120x^3y^7.
To find the seventh term of the expansion of (x + y)^10, we can use the binomial theorem. The binomial theorem states that the expansion of (x + y)^n can be written as:
(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) * x^0 * y^n
where C(n, r) represents the binomial coefficient, also known as "n choose r", which calculates the number of combinations of choosing r items from a set of n items.
In this case, we want to find the seventh term, which means we need to find the term with x raised to the power of 10-7 = 3 and y raised to the power of 7. The term will be multiplied by C(10, 3) = 10! / (3!(10-3)!) = 120.
So, the seventh term is given by:
C(10, 3) * x^3 * y^7 = 120 * x^3 * y^7
Therefore, the seventh term of the expansion (x + y)^10 is 120x^3y^7.
using Pascal's Triangle for the coefficient
210 x^3 y^7
or ... where p is the power and n is the number of the term
... use the combination for the coefficient ... p C (n-1)