If we wish to expand (x + y) 8 , what is the coefficient of x 5 y 3 ? what is the coefficient of x 3 y 5 ?

We use the binomial theorem to answer this question. Suppose we have a trinomial (a + b)ⁿ, we can determine any term to be:

[n!/(n-r)!r!] a^(r) b^(n-r)

a.) For x⁵y³, the variables are: x=a and y=b. We already know the exponents of the variables. So, we equate this with the form of the binomial theorem.

r = 5
n - r = 3
Solving for n,
n = 3 + 5 = 8

Therefore, the coefficient is equal to:
Coefficient = n!/(n-r)!r! = 8!/(8-5)!8! = 56

b.) For x³y⁵, the variables are: x=a and y=b. We already know the exponents of the variables. So, we equate this with the form of the binomial theorem.

r = 3
n - r = 5
Solving for n,
n = 5 + 3 = 8

Therefore, the coefficient is equal to:
Coefficient = n!/(n-r)!r! = 8!/(8-3)!8! = 56

The earths atmosphere is made up of nitrogen and oxygensolids turning into gases is called sublimation.