Question

Find the expansion of

a) (x + y)^6

b) (x + y)^4

31) What is the coefficient of x^12 y^13 in the expansion of (x + y)^25?

32) What is the coefficient of x9 in (2 − x)19?

33) What is the coefficient of x^101 y^99 in the expansion of (2x − 3y)^200?

Answer 1

a)

$$(x+y)^6=x^6+6x^5y+15x^4y^2+20x^3y^3+15x^2y^4$$

$$+6xy^5+y^6$$

b)

$$(x+y)^4=x^4+4x^3y+6x^2y^2+4xy^3+y^4$$

31)

$$\dbinom{25}{12}=\dfrac{25!}{12!(25-12)!}=5200300$$

32)

$$(-1)^9\dbinom{19}{10}(2)^{10}=-\dfrac{19!}{10!(19-10)!}(1024)$$

$$=-92378(1024)=-94595072$$

33)

$$(-1)^{99}\dbinom{200}{101}(2)^{101}(3)^{99}$$

$$=-\dfrac{200!}{101!(200-101)!}(2)^{101}(3)^{99}$$