Find the coefi�cient of x^18 y^32 z^25 in (x + y + z)^75? How many monomials appear in the expansion of (x + y + z)^75?
To find the coefficient of a specific term in a binomial expansion, you can use the binomial theorem formula or Pascal's Triangle. In this case, we can use the binomial theorem formula.
The binomial theorem states that for any positive integers a, b, and n, the term of the form (x + y)^n is given by the formula:
C(n, r) * x^(n-r) * y^r
where C(n, r) represents the binomial coefficient and is calculated using the formula:
C(n, r) = n! / (r! * (n-r)!)
In our case, we need to find the coefficient of x^18 y^32 z^25 in the expansion of (x + y + z)^75. Since the sum of the exponents of x, y, and z must be equal to 75, we can set up the following equation:
18 + 32 + 25 = 75
To find the coefficient, we need to calculate the binomial coefficient using the formula, substituting n = 75 and r = 18:
C(75, 18) = 75! / (18! * (75-18)!)
To calculate this, you can use a calculator or computer software capable of handling large factorials. Once you calculate the binomial coefficient, you will have the coefficient of the term x^18 y^32 z^25.
Now, let's determine the number of monomials in the expansion of (x + y + z)^75. Each monomial in the expansion corresponds to a term of the form x^a y^b z^c. To find the number of monomials, we need to find all possible combinations of a, b, and c that satisfy the condition a + b + c = 75.
One way to find this is by using the stars and bars method. Imagine there are 75 stars representing the total exponent, and we need to separate them into three groups corresponding to the exponents of x, y, and z. We can place two bars to divide the stars into three groups. The positions of the bars determine the values of a, b, and c.
Thus, the number of monomials is equal to the number of ways we can arrange 2 bars among 75 stars, which can be calculated using the formula:
C(75 + 2, 2) = (75 + 2)! / (2! * (75 + 2 - 2)!)
Again, you can use a calculator or computer software to calculate this.