I need to find the coefficient of x^18 y^32 in (x+y)^50.
I understand that we have to use the binomial theorem. I know how to
find the coefficient for example x^18 but here we have "x" as well as
"y" which I have no idea about.
general term(r+1)
= C(50,r)(x^(50-r) y^r
comparing this with ? x^18 y^32 we can see that r = 32
so we want C(50,32) which is 1.80535 x 10^13
How many monomials appear in the expansion of (x + y + z)^75?
To find the coefficient of a specific term in the expansion of a binomial expression, like (x+y)^50, you can use the binomial theorem. In this case, we need to find the coefficient of x^18 y^32.
The binomial theorem states that for any positive integers n and k,
(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,k)x^(n-k)y^k + ... + C(n,n)x^0 y^n
where C(n,k) represents the binomial coefficient, which is given by the formula:
C(n,k) = n! / (k!(n-k)!)
In this case, n = 50, so we need to find the term with k = 32 in the expansion of (x+y)^50.
To find the coefficient of x^18 y^32, we need to find the value of C(50,32). The binomial coefficient C(50,32) can be calculated using the factorial formula:
C(50,32) = 50! / (32!(50-32)!)
To simplify the calculation, we can use the property that n! = n(n-1)!. Therefore,
C(50,32) = 50 * 49 * 48 * ... * 19 / (32 * 31 * 30 * ... * 1)
Now you need to calculate the numerator and the denominator separately and then divide them to find the coefficient of x^18 y^32 in (x+y)^50.