Binomial theorem

Question

(2n-3m)^4 , 4th term

(4a + 2b )^8 , 5th term

Answer 1

(2 n - 3m)^4

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(4 choose 0) = 1

(4 choose 1) = 4

(4 choose 2) = 4 * 3 / 2! = 6

(4 choose 3) = 4

(4 choose 4) = 1

(A + B )^4 = A^4 + 4 A^3 B + 6 A^2 B^2 + 4 A B^3 + B^4

A=2n , B = -3m

(2n)^4 + 4( 2n)^3 (-3m) + 6 ( 2m)^2 (-3n)^2 + 4 (2n) (-3m)^3 + (-3m)^4 =

16n^4 + 4( 8n^3)(-3m) + 6 ( 4m^2) (9n^2) + 4(2n)(-27m^3) + 81 m^4 =

16n^4 - 96 n^3 m + 216 m^2 n^2 - 216 n m^3 + 81 m^4

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(8 choose 0) = 1

(8 choose 1) = 8

(8 choose 2) = 8*7/2! = 28

(8 choose 3) = 8*7*6/3! = 56

(8 choose 4) = 8*7*6*5/4! = 8*7*6*5/(4*3*2) = 7*6*5/3 = 70

(8 choose 5) = (8 choose 3) = 56

(8 choose 6) = (8 choose 2) = 28

(8 choose 7) = (8 choose 1) = 8

(8 choose 8) = 1

(X + Y)^8 = X^8 + 8 x^7 y + 28 x^6 y^2 + 56 x^5 y^3 + 70 x^4 y^4 + 56 x^3 y^5 + 28 x^2 y^6 + 8 x y^7 + y^8

X=4a Y= 2b

(4a)^8 + 8 (4a)^7 (2b) + 28 (4a)^6 (2b)^2 + 56 (4a)^5 (2b)^3 + 70 (4a)^4 (2b)^4 + 56 (4a)^3 (2b)^5 + 28 (4a)^2 (2b)^6 + 8 (4a) (2b)^7 + (2b)^8=

65536 a^8 + 262144 a^7 b + 458752 a^6 b^2 + 458752 a^5 b^3 + 286720 a^4 b^4 + 114688 a^3 b^5 + 28672 a^2 b^6 + 4096 a b^7 + 256 b^8

Answer 2

1) According to binomial theorm coefficients of the 5 terms are 1,4,6,4,1 so 4th term is

4(2n)^1(-3m)^3= 4(2n)(-27m^3)= -216nm^3

2) coefficients for the 9 terms are 1,8,28,56,70,56,28,8,1 so 5th term is

70(4a)^4(2b)^4= 70(256a^4)(16b^4) = 286720a^4b^4