state whether or not a constant will appear in the expansion of (x+y)^11
what does this mean? please help
in the expansion of
(x+y)^11 we get
x^11 + 11x^10y + 55x^9y^2 + ... + 11xy^10 + y^11
each term will have some combination of x and y so that the exponents add up to 11
A constant would have no variables, which cannot happen in the above question.
To determine whether a constant term will appear in the expansion of (x+y)^11, we can use the binomial theorem. The binomial theorem states that for any positive integer n, the expansion of (a+b)^n can be written as the sum of terms in the form of C(n,k) * a^(n-k) * b^k, where C(n,k) represents the binomial coefficient "n choose k".
In the case of (x+y)^11, the constant term will appear when k is equal to n, which means when all the powers of x and y cancel out. Here, n is 11, so we need to find C(11,k) * x^(11-k) * y^k such that k is 11.
Using the formula for binomial coefficient C(n,k) = n! / (k!(n-k)!), we can calculate C(11,11) as follows:
C(11,11) = 11! / (11!(11-11)!) = 11! / (11!(0)!) = 11! / (11! * 1) = 1
Thus, C(11,11) is equal to 1. When k is equal to 11, the term becomes 1 * x^(11-11) * y^11 = y^11.
Therefore, the term y^11 is the constant term in the expansion of (x+y)^11, and yes, a constant will appear in the expansion.