Use the binomial theorem to write the binomial expansion of

Use the binomial theorem to write the binomial expansion of (x - 3y^2)^5. 11. Use the binomial theorem to find the 12th term in the binomial expansion of (-2Squarerootx + 1/2y)^19. 12. Find the 8th number in the 11 throw (n = 11) of Pascal\'s triangle. 13. Show that the sum of the 34th and 35th numbers in the 117th row of Pascal\'s triangle is equal to the 85th number in the 118th row of the triangle.

Solution

11) (x -3y^2)^5

Binoimal expansion formula : (a+b)^n = nC0(a)^(n-0)b^0 +nC1a^(n-1)b^1 + nC2 a^(n-2)b^2........

(x -3y^2)^5 = 5C0(x)^(5-0)(-3y^2)^0 +5C1(x)^(5-1)(-3y^2)^(1) +5C2x^(5-2)(-3y^2)^2 +5C3x^(5-3)(-3y^2)^3 +5C4x^(5-4)(-3y^2)^4 + 5C5(x)^(5-5)(-3y^2)5

x515x4y2+90x3y4270x2y6+405xy8243y^101

12 ) ( -2sqrtx + y/2)^19

12 th term of binomial expansion : 19C11(-2sqrtx)^(19-11)(y/2)^11

we know : 19C11 = 75582

= (37791x^4y^11)/4