Question

Consider the product (x + y + z)2. If the expression is multiplied out and like terms collected, the result is: x2 + y2 + z2 + 2xy + 2yz + 2xz Suppose we do the same to the product (v + w + x + y + z)25 (a) What is the coefficient of the term v9w2x5y7z2? (b) How many different terms are there? (Two terms are the same if the degree of each variable is the same.)

Answer 1

Answer:

Part a: The coefficient of $v^9w^2x^5y^7z^2$ is $1.766 \times 10^{13}$

Part b: The number of terms are 23751.

Step-by-step explanation:

part a

From the given equation the

$(x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2zx$

So the coefficient of any term $x^ny^nz^n$ is given as  

$\dfrac{2!}{n!n!n!}$

Similarly for the generic equation of coefficient of the term $x_1^{r_1}x_2^{r_2}x_3^{r_3}......x_k^{r_k}$ in the equation of form $(x_1+x_2+x_3+x_4......x_k)^n$ is given as  

$\dfrac{n!}{r_1!r_2!r_3!....r_k!}$

So now the coefficient of $v^9w^2x^5y^7z^2$  is given as

$=\dfrac{25!}{9!2!5!7!2!}\\=1.766 \times 10^{13}$

The coefficient of $v^9w^2x^5y^7z^2$ is $1.766 \times 10^{13}$

Part b:

The number of terms is given as $\left (\ {{m+n-1} \atop {n}} \right )$

where m is the number of variables which are 5 here

n is the power which is 25 so the number of variables is given as

$\left (\ {{5+25-1} \atop {25}} \right )\\\left (\ {{29} \atop {25}} \right )\\\dfrac{29!}{25!4!}=23751$

So the number of terms are 23751.

Answer 2

Part(a): The coefficient of the term $v^{9}w^{2}x^{5}y^{7}z^{2}$ is $\frac{25!}{9!\times 2!\times 5!\times 7!\times 2!}$

Part(b): The number terms are 23751

Part(a):

Given,

The given expression is, $(x^{2}+y^{2}+z^{2})^{2}$

The number of coeffecient is given by,

$\frac{(9+2+5+7+2)!}{9!\times 2!\times 5!\times 7!\times 2!}} =\frac{25!}{9!\times 2!\times 5!\times 7!\times 2!}$

Part(b):

The given expression is, $(x^{2}+y^{2}+z^{2})^{2}$

Let, $v^{9}w^{2}x^{5}y^{7}z^{2}$ be the term of the expansion $(v+w+x+y+z)^{25}$

Therefore, $a+b+c+d+e=25$

The total number of different terms is equal to the number of ways we can distribute 25 items among 5 people.

Then the number of different terms are,

$^{25+5-1}C_{5-1}=^{29}C_{4}\\=23751$

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