Determine the coefficient of
wx^3y^2z^2 in (2w−x+y−2z)^8
Formula for Coefficients in a Trinomial Expansions is
\frac{n!}{r!}
n = power of the given expansion = 8
r = power of the asked coefficient
here it is asked for the coefficient of wx^{3}y^{2}z^{2}
their powers
w=1, x=3, y=2, z=2
so r = 1!* 3! * 2!* 2!
\Rightarrow\frac{8!}{1!* 3!*2!* 2!}
= 1680
\therefore coefficient of wx^{3}y^{2}z^{2} is 1680
To determine the coefficient of wx^3y^2z^2 in the expression (2w−x+y−2z)^8, we can use the binomial theorem.
The binomial theorem states that for any positive integer n, we can expand the expression (a + b)^n using the formula:
(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) * a^0 * b^n,
where C(n, k) represents the binomial coefficient, given by the formula:
C(n, k) = n! / (k! * (n-k)!),
and n! denotes the factorial of n.
In our case, the expression is (2w−x+y−2z)^8, with a = 2w−x and b = y−2z. Therefore, we need to find the term that contains wx^3y^2z^2.
Let's break down the term wx^3y^2z^2:
- The coefficient of wx^3y^2z^2 is 1.
- The exponent of w is 1, so we need to have w to the power of 1, which means it appears once in our term.
- The exponent of x is 3, so we need to have x to the power of 3, which means it appears three times in our term.
- The exponent of y is 2, so we need to have y to the power of 2, which means it appears twice in our term.
- The exponent of z is 2, so we need to have z to the power of 2, which means it appears twice in our term.
Now we can focus on finding the specific term in the expansion.
Using the binomial theorem, we can see that the coefficient of the term containing wx^3y^2z^2 is given by the binomial coefficient for the respective exponents. In this case, it is C(8, 3) * (2w−x)^3 * (y−2z)^2.
To find C(8, 3), we use the formula for the binomial coefficient:
C(8, 3) = 8! / (3! * (8-3)!) = 8! / (3! * 5!).
Now, we can calculate C(8, 3):
C(8, 3) = (8 * 7 * 6 * 5!) / (3 * 2 * 1 * 5!) = 8 * 7 * 6 / (3 * 2 * 1) = 56.
Therefore, the coefficient of wx^3y^2z^2 in (2w−x+y−2z)^8 is 56.
By applying the binomial theorem and calculating the appropriate binomial coefficient, we were able to determine the coefficient of wx^3y^2z^2 in the given expression.